TA 



• HANDBOOK •: 

OF 

MATHEMATiCS 

FOR 

ENGINEERS . 

l,A.WArERBURy 



WITH 



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WORKS OF THE LATE 

L. A. WATERBURY 

PUBLISHED BY 

JOHN WILEY & SONS. Inc. 



Stresses in Structural Steel 
Angles with Special Tables 

V + 77 pages. 5| x 8. Illus- 
trated. $1.25 net. 

Cement Laboratory Manual 

A Manual of Instructions for 
the Use of Students in Cement 
Laboratory Practice. vii + 
122 pages. 5 by 7^. 28 fig- 
ures. Cloth, $1.00 net. 

A Vest=pocket Handbook of 
Mathematics for Engineers 
with Tables 

Third Edition, Enlarged, xiv 
+ 278 pages. Fully illustrated. 
Flexible binding, $1 50 net. 



A VEST-POCKET 
HANDBOOK 

OF 

MATHEMATICS 

FOR 

ENGINEERS 



of Illinois 
AND 

H. H. HIGBIE 

Professor of Electrical Engineering, University 
of Michigan 



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BY i< 

L. A. ^ATERBURY ^ 

La^e Professor of Civil and Architectural Engineering, ^ 
University of Arizona *" 

WITH SPECIAL SECTIONS 
BY 

G. A. GOODENOUGH ^ i 

Professor of Thermodynamics, University 



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THIRD EDITION ENLARGED 
THIRD PRINTING CORRECTED x 



NEW YORK ^ 

JOHN WILEY AND SONS, Inc. 

London: CHAPMAN & HALL, Limited ^ = 






Copyright, 1908, 1909 

BY 

L. A. WATERBURY 



Copyright, 1919 

BY 

ETHEL CLAIRE MILLER WATERBURY 



Oh 



X 



Stanbopc iprcss 

F, H.GILSON COMPANY 
BOSTON, U.S.A. 



2-24 



PREFACE TO THIRD 
EDITION 






The former editions of this handbook have 
been so well received that the publishers, 
Messrs. John Wiley and Sons, Inc., suggested 
the possibility of increasing its usefulness by 
the addition of material relating to thermo- 
dynamics and to electrical engineering. For 
the preparation of a section on heat engi- 
neering, Professor G. A. Goodenough, of the 
University of Illinois, was selected, while 
Professor H. H. Higbie, of the University of 
Michigan, was chosen to prepare a section on 
electrical engineering. These two new sec- 
tions and their related tables constitute the 
principal addition which has been made to 
the former edition. 

L. A. W. 

NiTRO, W. Va., May, 1918. 



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PREFACE TO SECOND 
EDITION. 

In preparing the second edition, the errors 
which have been discovered in the previous 
edition have been corrected, revisions and alter- 
ations have been made throughout the work, 
and new material has been added, including 
sections on hydraulics and reinforced concrete, 
and a table of conversion factors. 

L. A. W. 

Urbana, III., April, 1915. 



PREFACE. 



This handoook is intended -as a reference 



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book, for the ust of those who have studied < o 



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or are studying tie branches of mathematics "^ o 
usually taught in engineering courses. It is 
not intended for a text book, and does not, ^ « 
therefore, attempt to prove many of the 
formulae which are given. 

Most of the material in this book was 
obtained from the following sources: algebra 
from Hall & Knight's Algebra (Macmillan 
Co.); trigonometry from Bowser's Trig- 
onometry; analytic geometry from Candy's 
Analytic Geometry; calculus from Taylor's 
Differential and Integral Calculus; theoret- 
ical mechanics from Church's Mechanics of 
Engineering; and mechanics of materials 
from Merriman's Mechanics of Materials; 
to all of which the writer is very much in- 
debted and from all these Authors he has ^ 
received permission to use the material. The \ < 
reader is referred to these works for the proof \ 
and explanation of the various formulae. 

L. A. W. 
Tucson, Ariz., March, 1908. 

PREFACE TO FIRST EDITION 
WITH TABLES 

In this edition tables of logarithms of num- 
bers, natural and logarithmic sines and co- 
sines, and natural and logarithmic tangents 
and cotangents have been added to facilitate 
the solution of problems. 

L. A. W, 

TucgON, Ariz., September, 1909, 



r I 

D O 

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E S 



- (£ 



CONTENTS. i 



QC 



PAGE •" i 

Algebra 1 

Exponents and Logarithms 1 o > 

Proportion 2 >& 

Progression, etc 3 zg 

Quadratic Equations 5 " 

Cubic Equations 5 ^ 

Higher Equations 6 - ^ 

Graphical Solution of Equations ... 7^3 

J- < 

Trigonometry 8 5 

Plane Triangles 11 

Spherical Triangles 12 m 

Hyperbolic Functions 13 ^ 

-J 

Analytic Geometry 14 ^ 

Transformation of Coordinates ... 14 -.-^ 

The Straight Line 16 S S 

The Circle 17 ^ < 

The Parabola 18 ^ S 

The Ellipse 18 E ^ 

The Hyperbola 19 

The Cycloid 20 j | 

Miscellaneous Curves 20 ^ £ 

SoUds 21 3 < 

J s 
E a. 

Differential Calculus 23 

Integral Calculus 26 S 

Theoretical Mechanics 39 a 

Q 

Notation 39 i- < 

statics: 

Equilibrium of Forces 40 

Centroid of Parallel Forces .... 41 S^ c 

Center of Gravity 41 i^ "- ^ 

Moment of Inertia of an Area ... 43 f 
▼u < "i 



VIU CONTENTS 

PAGE 

Moment of Inertia of a Mass ... 44 

Product of Inertia 44 

Radius of Gyration 44 

Transformation Formulae 44 

Ellipsoid of Inertia 45 

Circle of Inertia 45 



Dynamics: 

Velocity and Acceleration .... 46 

Uniformly Accelerated Motion ... 46 

Falling Bodies 46 

Force and Acceleration 47 

Direct Central Impact 47 

Virtual Velocities 47 

Curvilinear Motion of a Point ... 48 

ProjectUes 48 

Translation of a Rigid Body .... 49 

Rotation of a Rigid Body 49 

Precessional Rotation 51 

Center of Percussion or Oscillation . 51 

Pendulum 52 

Work, Energy, and Power 52 

Friction 52 

Friction of Belt 53 

Mechanics of Materials 54 

Physical Properties of Materials ... 54 

Notation 55 

Direct Stress 55 

Eccentric Loads 56 

Equation of Neutral Axis 59 

Kernel or Core-section 59 

Section Modulus Polygons 60 

Diagonal Stresses . 62 

Thin Pipes, Cylinders, and Spheres . . 63 

Riveted Joints 64 

Beams 65 

Simple Beam, Uniform Load ... 69 

Simple Beam, Load at Center ... 70 



CONTENTS 


IX 




PAGE 


Simple Beam, Unsymmetrical Load 


. 70 


Simple Beam, Several Loads . . . 


71 


Cantilever Beam, Uniform Load . 


72 


Cantilever Beam, Load at End 


73 


Fixed Beam, Uniform Load . . . 


73 


Fixed Beam, Load at Center . . 


74 


Fixed Beam, Load off Center . . 


76 


Continuous Beam, Uniform Load . 


77 


Stmts and Columns: 




Euler's Formula 


80 


Rankine's Formula 


81 


Ritter's Formula 


81 


The Straight Line Formula . . . 


82 


Eccentrically Loaded Columns . . 


. 84 



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Torsion 



85 



Ellipsoid of Stress 



86 



Reinforced Concrete: 

Notation 

Columns with Longitudinal Steel . 
Colmnns with Spiral Steel .... 

Beams in General . ." 

Rectangular Beams 

T-Beams 

Beams Reinforced for Compression 
Flat Slab Floors 



86 
87 
87 
87 
88 
88 
88 
89 



Hydraulics: 

Notation .... 
Static Pressure 
Center of Pressure 



92 
92 
93 



Weirs: 

Francis Formula 

Fteley and Stearns' Formula 

Bazin's Formula 

Hamilton Smith's Formula . 
Trapezoidal Weirs (Cippoletti) 



93 
93 
93 
94 
94 



X CONTENTS 

PAGE 

Triangular Weirs 94 

Submerged Weirs 94 

Orifices and Jets: 

Discharge 95 

Force and Energy 96 

Moving Plates 96 

Flow in Pipes: 

Long Pipes 97 

Fanning' s Formula 97 

Chezy's Formula 98 

. Flamant's Formula 98 

Various Losses of Head 98 

Equivalent Pipe Lengths 99 

Bernoulli's Theorem 99 

Flow in Channels: 

Chezy's Formula 99 

Kutter's Formula 100 

Bazin's Formula 100 

Relations of Velocities 101 

Hydraulic Grade Line 101 

Heat Engineering: 

Elements of Thermodynamics: 

Notation 102 

Fundamental Equations: Definitions 102 

Perfect Gases 103 

Values of B, Cp, c^, and k for Gases: 

Special Changes of State 104 

Saturated and Superheated Steam: 

Notation 106 

Fundamental Relations 106 

Equations for Superheated Steam . 107 

Tables of the Properties of Steam . 107 
Changes of Stat3 in Steam and Water 

Mixtures 108 



CONTENTS XI 

PAGE 

Flow of Compressible Fluids: 

Equation of Continuity 108 

Equation of Energy 109 

Discharge through Orifices 109 

Diverging Nozzles Ill 

Flow of Gases and Vapors in Mains. 113 

The Steam Engine: 

Ideal Rankine Cycle 114 

Efficiency of the Actual Engine . . 116 

Steam Boilers 116 

Condenser 117 

Internal Combustion Engines: 

Otto Cycle 117 

Diesel Cycle 119 

Air Compression: 

Compound Compression 120 

Refrigeration : 

Air as the Medium 121 

Vapor as the Medium 121 

Electrical Engineering Formul2E2: 

Notation 124 

Magnetic Forces and Fields: 

Field Due to a Pole at a Point . . . 127 

Field Due to Current in Straight Con- 
ductor 127 

Force on Conductor Due to Current 

and Field 127 

Law of the Magnetic Circuit . . . 128 

Magnetically Induced Electromotive 

Force 128 

Inductance of an Electric Circuit: 

General 129 

Self-Inductance of a Transmission 

Line 130 

Mutual Inductance of Two Electric 

Circuits 130 

Energy Stored in Magnetic Field . . . 130 



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XU CONTENTS 

PAGE 

Power Dissipated in Magnetic Circuit . 131 
Condensers and Electrostatics .... 131 
Circuits Carrying Direct Current (Un- 
varying) 133 

Direct-current Machines 138 

Growth and Decay of Current in In- 
ductive Circuit 139 

Harmonic Alternating Current .... 140 

Three-phase Circuits: 

Star or Wye Connection 145 

Delta or Mesh Connection .... 147 

Power in Three-phase Systems . . . 148 

Magnetization Curves for Electrical 

Steels 150 

Hysteresis Loss 150 

Eddy-current Loss 151 

Dielectric Constants 151 

Resistivity (po) at 0° Cent 152 

Wire Table for Round Wires .... 153 

Tables 

I. Logarithms of Numbers 156 

II. Logarithmic Sines and Cosines . . 186 

III. Logarithmic Tangents and Cotan- 

gents 204 

IV. Natural Sines and Cosines . . . 222 
V. Natural Tangents and Cotangents . 240 

VI. Conversion Factors 258 

'VII. Properties of Saturated Steam 

(Goodenough) 262 

VIII. Pressure-entropy Table for Steam 

(Goodenough) 266 



GREEK LETTERS. 



O H 

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A 


a 


Alpha 


N 


V 


Nu 


B 


^ 


Beta 


s 


i 


Xi 


r 


y 


Gamma 





o 


Omicron 


A 


8 


Delta 


n 


IT 


Pi 


E 


€ 


Epsilon 


p 


P 


Rho 


Z 


r 


Zeta 


2 


<T 


? Sigma 


H 


V 


Eta 


T 


T 


Tau 


e 


d^ 


^ Theta 


T 


V 


Upsilon 


I 


L 


Iota 


^ 


<t> 


Phi 


K 


K 


Kappa 


X 


X 


Chi 


A 


X 


Lambda 


^ 


^ 


Psi 


M 


M 


Mu 


12 


03 


Omega 



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ALGEBRA. 



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EXPONENTS AND LOGARITHMS 

If a** = 6, m = loga 6. a'^'a'^ = a"*+^, 
.*. log (x • 2/) = log X + log 2/. a~^ -=- a*» = a***"^, 
.*. log (a; 4- 2/) = log x — log 2/. (a'»)2 = 

o"* . a"* = a2 »«, /. log x2 = 2 • log a:. 
ifl/^)'' = a"* • ", .-. log x" = 71 • log X. 
aO = l, .-. log.(l)=0. 

For common logarithms the base is 10; 
log 10 = 1, log 100 = 2, log 1000 = 3, etc., or 
for any number between 1 and 10, the loga- 
rithm will have a value between and 1, and may 
be found in a table of logarithms. The values 
of the logarithm of any number may be ob- 
tained by adding the proper integer to the 
proper value obtained from the tables. For 
example, 

log (451.7) = log (4.517 X 100) 

= log 4.517 + log 100 

= 0.65485 + 2 

= 2.65485. 

It may be observed that the integral part of 
the logarithm, called the characteristic, in- 
dicates the location of the decimal point of the 
number; and that the decimal portion of the 
logarithm, called the mantissa, determines the 
sequence of significant figures. 

For a number less than unity, the logarithm 
is negative, but since the tables contain only 
positive values, the logarithm for such a num- 
ber is ordinarily used in the form of a positive 
mantissa with a negative characteristic. For 
the purpose of involution or evolution the 
X 



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ALGEBRA 



logarithm may well be used in the negative 
form. For example, 

log (0.04517) = log (4.517 -^ 100) 

= log 4.517 -log 100 

= +0.65485-2 
(or =+8.65485-10) 

= - 1.34515. 
(The logarithm is usually written 2.65485.) 
log (0.04517)1.6 = (1.6)- log (0.04517) 

= (1.6) ( - 1.34515) 

= -2.15224 

= +0.84776-3 

= log (0.007043), 
.-. (0.04517)1.6 = 0.007043. 

The base of the natural system of logarithms 



IS 



'+'+Q+il+a 



-t 



= 2.7182818284. 



The cologarithm of a nimiber is the loga- 
rithm of its reciprocal. Log (~) = ~ log ^. 

To transform a logarithm from base e to 
base 10, multiply by logio e. 

Logio 6 = 0.43429448. 
Logg 10 = 2.30258509. 
1 ' 









Jjugio e - 


log^lO 










PROPORTION. 


If a 


:6: 


:c 


:d, 












a c 
b-d 


b 
or - = 
a 


d 

s— t 

c 








ad = bc. 


a + & 
b 


c + rf 
d 




o- 


-b 


c-d 


c + 6 


c+rf 



a — b c-^d 



ALGEBRA 
ARITHMETICAL PROGRESSION. 



a, a + d, a + 2d, . . . 




Last term, L = a + (n — l) d. 
Sum of terms, 


^g 


S = ^(a + L)=^[2a + (n-l)d]. 


H 

> Li 


GEOMETRICAL PROGRESSION. 


<s 


a, ar, ar^, ar^, . . . 


<co 


Last term, L = ar^'^. 
Geometric mean, M = ^/a6. 

Sum. 5 = «('--" 


1- 3 
Z -I 

q: o 

UJ -1 

Q 


aCl-r**) rL-a 


. « 



1-r r-1 

For an infinite geometrical series, the sum 
a 



to infinity is S = 



1-r 



HARMONIC PROGRESSION. rS 

D O 

a, b, c are in harmonic progression if r s 



a a — b 



n-p \n 



3< 

7 a. 



or if -» T» - are in arithmetical progression. I h 

a h c i< 

u z 
E u. 

PERMUTATIONS AND COMBINATIONS. 

ab and fea are two permutations but only S 

one combination. > ^ 

The number of permutations possible of g 

n things taken r at a time is 

^P^=n(n-1) (n-2) . . . (n-r + 1). 

(tri = lX2X3X4 . . . Xn). 



5i 



z: z 



ALGEBRA 



BINOMIAL THEOREM. 



|2 



• a»t~2 . 62 



n'(n-l) (n-2) 
13 



an-2 . 53 



SERIES. 

1. An infinite series in which the terms 
alternately positive and negative is convergent 
if each term is numerically less than the pre- 
ceding term. 

2. An infinite series in which all the terms are 
of the same sign is divergent if each term is 
greater than some finite quantity, however 
small. 

3. An infinite series is convergent if from 
and after some fixed term the ratio of each 
term to the preceding term is numerically less 
than unity. 

4. An infinite series in which all the terms 
are of the same sign is divergent if from and 
after some fixed term the ratio of each term 
to the preceding term is greater than unity, 
or is equal to unity. 

5. If there are two infinite series in each 
of which all the terms are positive, and if the 
ratio of the corresponding terms in the two 
series is always finite, the two series are both 
convergent, or both divergent. 



are S 





DETERMINANTS. 


= 0162 - aih. 

0,2 ^2 




ai 61 ci 


= Ol • ?>2 • C3 -{- 




a2 62 C2 


02 • 63 • Ci + 




as 63 C3 


^3 • 61 • C2 


- ai . 63 . C2 




■ 0^ • 61 • c 


} - as • 6a . ci. 



ALGEBRA 



If 



then 



aix + biy + ciz + di = 0, 
a^x -\-h2y-\- c^z + (^2 = 0, 
a^x + 63^ + C32 + da = 0, 






= -1 



6icidi 




aicic^i 




ai6idi 




ai6ici 


62C2(^2 




a2C2<?2 




a^^di 




0262C2 


feacsc^s 




ascsda 




a^zdz 




aa^aca 



QUADRATIC EQUATIONS. 

ax2 + 6a; + c = 



2a 



or 



CUBIC EQUATIONS* 


Hrst Form. 


x3 + 6a: + c = 0. 




2/«-hc2/3-|,=0. 


C , ,/c2 , 63. 



(1) 

(2) 
(3) 
(4) 



from which x may be obtained by substituting 
the value of y in equation (2). 
Second Form. 

a:3 + aa;2 + c = 0. (5) 

Let x = l/z (6) 



or 



c c 



(7) 



which may be solved by equations (1) to (4) 
and the value of x may then be obtained by 
equation (6). 
Third Form. 

a;3 + ax2 + 6x + c = 0. (8) 

Let x = z-%, (9) 

o 

* The equations here used follow the method 
given in Wells' University Algebra. 



JL 0= 

g Li 

H O 

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6 



ALGEBRA 



which, when substituted in equation (8), will 
give an equation of the first form, the solution 
of which will give the value of 2, from which x 
may be obtained by equation (9). 

HIGHER EQUATIONS* 

For higher algebraic equations, an approxi- 
mate numerical solution can be obtained by the 
method of double position, as follows: 

/(x) = a:^ + aa:«-i + 6XW-2 . . . = 0. (1) 

By trial find two numbers one of which when 
substituted for x makes f{x) positive, and the 
other when substituted for x makes fix) nega- 
tive. Let a and 6 be the two numbers, and 
let A and B be the respective corresponding 
values of /(x) . Then, approximately, 

A : 5 = (x - a) : (x - 6) (2) 

A{h-d) 



or 



x = a-\-- 



A-B 



(3) 



GRAPHICAL SOLUTION OF EQUATIONS. 

To determine the value of x in any equation, 
f(x) =0, let 2/ = f{x) and compute the values 
of y for a nimaber of assumed values of x. 
Using the values of x and y as coordinates, plot 
the graph of the equation, y = /(re) , from which 
the value of x which will make f(x) become 
zero can be observed. 

For two simultaneous equations, involving 
two unknowns, the graph of each equation 
may be plotted with reference to one set of 
axes. If the two graphs intersect, the points 
of intersection will have coordinates which are 
the values of the two unknowns. If the graphs 
can not be made to intersect, there are no real 
values of x and y which are common to both 
equations. 

* See Wells* University Algebra. 



ALGEBRA 



For any equation, y = f{x) , the logarithms ^ 

of X and y may be plotted instead of the quan- o h 

tities themselves, producing the logarithmic p g 

graph of the equation. Logarithmic graphs ^ 

are particularly useful for equations of the — 

form, y = ax*, for which the graphs are straight p ^ 

lines. J s 



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4 



TRIGONOMETRY. 







Radius - 1, 


Q F AD^ci^Q 


J 


^ 

N 


5 OA = cos 6. 
CD = tan d. 
EF = cot 0. 


AC 
Fig. 1 


OD = sec 0. 
OF = cosec 


AC = vers = 1 - cos 0. 


BG = covers = 1 - sin 0. 


/. sin^ 
tan0 = -' 

COS0 


sin2 + cos2 = l. 


sec20 = l + tan20. 


cosec2 = 1 + cot2 d. 


exsec ^ = sec — 1. 


For 6 in radians, 


sin< 


? = 


^ 03 fl5 ^ 



cos0=l-2+g-[6+ • •• 



X ., z, . ^' I 2-05 1707 

tan0 = 0+3 +37^+^:3;^:^ + 

8 



TRIGONOMETRY 



9 



sin (A + S) = 
sin (A - B) = 
cos (A + B) = 
cos {A-B) = 

tan (A + 5) = 

tan (A - 5) = 

sin 2 A = 
cos 2 A = 



tan 2 A 

tan(|) 

sin 3 A 
cos 3 A 

tan 3 A 

sin A + sin 5 i 
sin A — sin B = 
cos A + cos B ■■ 
cos A -- cos B - 



sin A • cos B + cos A • sin B. 

sin A • cos B — cos A • sin B. 

cos A • cos B — sin A • sin B. 

cos A • cos B + sin A • sin B. 

tan A + tan B 
1 — tan A • tan B 

tan A — tan B 

1 + tan A • tan B 

2 • sin A • cos A. 

cos2 A — sin2 A 

2 cos2 A - 1 / 

l-2.sin2A. 

2 ' tan A 
' 1 - tan2 ji ' 



-V 



(1 — cosA). 



(1 + cosA). 



2 

1 — cos A 

sin A 

3 • sin A — 4 • sin' A, " 

4 cos3 A — 3 cos A. 

3 tan A — tan^ A 
1-3 tan2 A 

_ . A+B A-B 

2 • sin — ;; — • cos — - — « 



^ A+B . A-B 

2 cos — - — • sin 



o ^ + ^ 

2 cos — - — • cos 



2 
A-B 



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^ . A+B . A-B 

— < sin — - — • sin — :r — < 



" a 



10 



TRIGONOMETRY 







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1 


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1 


O) 


1 


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1-H 


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(D cc 


^^ 


<I> 


o 


02 


a> 


<D 


DO 




02 


03 O 




02 




O 


02 


03 


O 




O 


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O 


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ti o 




-»^ 


^ 


u 


.4,9 


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+ 

1—1 


B + 




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■f3 

o 
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+ 


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> 1' 


> 


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>' 1 


> 






«) 


1 '*' 








<a5 

<N 


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c 


rt 








fl <!, 


•as 


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^ 


^ 


C§ 


=« ^ 


1 


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+ 


1— t 


1 




+ 
1— t 


-^ A 

+ 5 




> 1 


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«£. 


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<:£> 




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1— 1 


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TRIGONOMETRY 

sin A + sinB _ tan ^ (A-{-B) 
sin A — sin B tan ^ {A — B) 

sin A + sin -g 
cos A -{- cos B 

sin A + sin B 
cos A — cos B 

sin A — sin .g 
cos A + cos 5 

sin A — sin B 
cos A — cos B 

cos A + cos B 
cos A — cos B 



11 



= tan HA + 5). 
= cot ^ (A -5). 
= tanHA--S). 
= cot|(A + ^). 



^p 



u. < 

a 



PLANE TRIANGLES. 

A + B + C= 180°. 



a 



sm A 
tan A 



cos A = 



b 


c 


sin 5 


sinC 


a • sin C 


6 — a • cos C 


62 + C2- 


-a2 




Fig. 2. 



2 6c 



or a2 = 62 + c2-2 6c«co84. 

a + b _ tanHA + .g) ^ 
a — 6 tan ^ {A — B) 

sin A + sin S + sin C 

ABC 

= 4 • cos -- • cos — • cos — • 

2i £d £1 

cos A + cos B + cos C 

,,. .A . B . C 

= 1 + 4 • sin — • sin - . sin -. 



f^ 



01 S 



tan A + tan B + tan C = tan A • tan B • tan C 



12 



where 



TRIGONOMETRY 

Area = 5 6 • c • sin A 
_ a2 gin B • sin C 
2 'sin A 

= Vs{s-a){s-b)is-c), 
s = 1 (a + 6 + c). 



SPHERICAL TRIANGLES. 

Center of sphere is at O. 

A 




Fig. 3. 

Right Spherical Triangles, Let C represent 
the right angle. 

cos c = cos a • cos b. 

sin 6 = sin 5 • sin c. 

tan a = cos J5 • tan c, 

tan a = tan A • sin 6, 

1 

cose 

COS A — sin -B • cos o. 



tan A • tan B = 



Oblique Spherical Triangles, 

sin a sin 6 sine , . 

- — 7 = -: — rfz = - — ^ = modulus, 
sin A sinjB sinC 

cos a = cos b • cos c + sin b • sin c • cos A. 

cos A = — cos B • cos C + sin B • sin C • cos a. 

cot a • sin 6 = cot A • sin C + cos C • cos 6. 

Let s = Ha + ^-fc), 



TRIGONOMETRY 



13 



then sin 



(t)=y 

-(t)=v/ 

(A\ _ A / sin (s — 6) ■ sin (s — c) 
2 / V sin s • sin (s — a) 



sin (s — 6) • sin (s — c) 
sin b • sin c 

sin s • sin (s — a) 



sin 5 • sin c 



sin 



cos 



tan 



cos S • cos (<S — A) 



sin B • sin C 



(D=v/: 

'-\ = 4 / cos (^ - B) ' co^ (>S - C) 



1) = v/= 



sin B • sin C 

cos S ' cos (iS» — A) 
cos iS-B)' cos (S-C) 



HYPERBOLIC FUNCTIONS. 

For -the equilateral hyperbola, x^ — y^ = a^, a 
series of functions can be obtained, analogous 
to the circular functions. 

Let X, y be the coordinates of any point P 

(Fig. 4) , let the radius OP = r, let v = the arc 

MP divided by r, 

and let OM = a. 

Then, 

sinh?z = y/a. 

co3hu = x/a. 

tanh u = y/x. 

coth u = x/y. 

sech u = a/x. 

cosech u = a/y. 




is 



LU -I 
U. < 



^3 
3B 



cc X 

o o 

ui ui 

X s 



S u. 
O 



< "^ 



ANALYTIC GEOMETRY. 



TRANSFORMATION OF COORDINATES. 

To transform an equation of a curve from 
one system of coordinates to another system, 
substitute for each 
variable its value in 
terms of variables of 
the new system. 

Rectangular Sys' 
tern. Old Axes Par- 
allel to New Axes. 
x' = x —h. 
y' = y —k, 
x = x'-jrh» 
y = y'+k. 

Rectangular System. Old Origin Coincident 
with New Origin, 





x* — x 'COS + 2/ 'sinO. 
v' = y * cos Q — x • sin d. 
x = x' 'Co^B — y' • sin Q, 

ysy'.cos0 + a;'»sin^, 
14 



ANALYTIC GEOMETRY 



15 



Rectangular System. Old Axes not Parallel 
to New Axes. Old Origin not Coincident 
with New Origin, 




Fig. 7. 
x' —{x — h) cos 9-{-(y — k)sm 6, 

y'= (y — k) cos d—{x — h) SIR e, 

x = x''COsd — y'»smd-\-h. 
y = y' • cosd -{■ x' 'smd + k. 



n 

>■ Ui 

n 



Polar and Rectangular Systems, 






Fig. 8. 



x = p' cos I 
2/ = p • sin 6 



P = Va;2 + 2/2. 

y 



tan6> 



cos » = 



Va;2 + 2/2 



< a> 

o o 

H z 

uj <: 

CE I 

o o 

X s 



CO 

m -J 
o < 

tu 2 
O 





2/ 


COt0 = 


Va;2 + 2/2 

X 

2/' 


8600 = 


Vx2 + 2/2 


a; 




Vx2+J/2 



<g 



HI 



16 



ANALYTIC GEOMETRY 



THE STRAIGHT LINE. 

Equations of Straight Line. An equation of 
the first degree containing but two variables 
can always be represented by a straight line. 

The equation of the straight line may as- 
sume the following forms, for the rectangular 
system of coordinates. 

Ax + By + C = .... (1) 
y = mx-\-k (2) 

in which m is the value of the tangent of the 
angle which the line makes with the X-axis, 
and k is the intercept on the F-axis between 
the line and the X-axis. 

y-y'^Aix-x') .... (3) 

in which x\ y' are the coordinates of a point 
of the line, and A is a constant. 



y -y 



ix-x') . 



(4) 



in which x', y' and x", y" are the coordinates 
of two points of the line. 
The volar equation of 
a straight line is 



p • cos (0 — a) = A; (5) 

where h is the length of 
the normal ON. 




Fig.9. 



Distance between Two Points. The distance 
between two points, x', y' and x", y'\ is equal 

to 

V(x'-x")2+(2/'-2/")2. 

The distance between two points, pi, 0i, 
and p2, 02. is equal to 



^Pi^-h 



P2^ 



2 Pi •P2' COS (^1 — ^2)* 



Angle between Two Lines. The angle be- 
tween two lines, y=m'x+k' and y=m"x-\-k"f 



ANALYTIC GEOMETRY 



17 



is the difference between the two angles 
whose tangents are m' and m". 

Area of Triangle. The area of the triangle 
whose vertices are {xy, y{), {x^, 2/2)* and (X3, 2/3) 
is equal to 

XiVil 



THE CIRCLE. 

The most general equation of the circle, 
for rectangular coordinates, is 

(x-a)2+(2/-6)2 = i22, 

in which a, b are the coordinates of the center 

of the circle, and R is the radius. 

The following are special equations of the 

circle for rectangular and polar systems of 

coordinates. 

Y 

P 



x2 + 2/2 =722. 

p = R. 





y^ = 2Rx -x2. 
p = 2R' cos 0, 




x^ = 2Ry- 2/2. 
p = 2R*amd, 

Y 
Fig. 12. 
Diameter of circle =2R = D. 
Circumference = 2 x 72 = tt D. 



Z _l 

lii Z) 

a. o 

m -I 

u- < 



^3 
23 



o < 

z a 

S LI. 

o 



Area 



4 



18 



ANALYTIC GEOMETRY 



THE PARABOLA. 

In Fig. 13, F is the focus, OF = OD = a, and 
L —L is the latus rectum = 4 a. 

FP 

Eccentricity, e — ^^ = 1. 

If the F-axis coincides with the directrix, 
DM, then 

2/2 = 4a (x — a). 

M N' 
Q 




O 

c 

Fig. 13. 

If the y-axis coincides with ON, passing 
through the vertex, then 

2/2 = 4 ax. 

For a symmetrical segment of a parabola, 
the area of the segment is exactly two-thirds 
of the area of the enclosing rectangle. 



THE ELLIPSE. 

?!4.l^= 1 
a2 "^ 62 ^* 




Fig. 14. 

F, F are foci. 

Eccentricity, e < 1. 

The area of the ellipse is equal to irdb. 



ANALYTIC GEOMETRY 



19 



THE HYPERBOLA. 




A — A = principal hyperbola. 
B — B = conjugate hyperbola. 
c — c = asymptote. 



Z -I 
CC O 






Principal hyperbola: — — r^ = 1. 



/p2 7/2 

Asymptotes: -5 — r^ 
Conjugate hyperbola: 



= 0. 

rc2 2/2 



a-* 



62 



When referred to the asymptotes as axes, 
the equations become: 



Principal hyperbola: xy = 
Conjugate hyperbola: xy = 



D - D is the di- 
rectrix. 



02+62 



F, F are foci. 
FP 



PQ 



= e>L 




For the equilateral 
hyperbola, a = 6, for 
which the equation of the principal hyperbola 
becomes x^ — 2/2 = a^. 



< m 
o o 
H z 



CO -J 

o < 



si 

S u. 

o 



x5 
a: 
o 



Zj z 






20 



ANALYTIC GEOMETRY 



THE CYCLOID. 




dA^ 



Fig. 17. 

2/ = a (1 — COS0), 
or x = a* vers~i (- ) — "^2 ay — y^. 

THE SPIRAL OF ARCHIMEDES. 

THE RECIPROCAL OR HYPERBOLIC 
SPIRAL. 



THE PARABOLIC SPIRAL. 
THE LITUUS OR TRUMPET. 

2 ^ 



THE LOGARITHMIC SPIRAL. 

log p = k'd. 

li k = 1, and logarithms to the base a are 
employed, then the equation may be written 



ANALYTIC GEOMETRY 



21 



THE CATENARY. 

THE CUBIC PARABOLA. 

y = kx^. 

THE SPHERE. 

R = radius, and D = diameter. 
For the origin at the center, 

x2 + 2/2 + 22=i22. 

Area of surface =4:Tr R'^ = Tr D^. 



Volume 



= 1x123 = 1^2)3. 



CONES. 

The equation of the cone generated by the 
line, z = mx + c, rotated about the Z-axis, is 



a;2+2/2 = 



m2 
1 



The volume of a cone is - Ah, where A is the 

d 

area of the base, and h is the altitude. 
OBLATE SPHEROIDS. 
The equation of the oblate spheroid gen- 

X^ ^2 

erated by the ellipse, -; + 7^ =1, rotated about 

its minor axis, is 

^ , ^2 22 ^ 
a2 "*" a2 "•" 62 ^• 

PROLATE SPHEROIDS. 

The equation of the prolate spheroid gener- 
a;2 ^2 
ated by the ellipse, t^ + -;; = 1, rotated about 



its major axis, is 



fe2 ^ 62 ^ 02 



t^ 



u. < 

u. o 

Q 



5i 



a CO 

< "J 



22 



ANALYTIC GEOMETRY 



HYPERBOLOIDS. 

The equation of the hyperboloid of one 
nappe, generated by the hyperbola, -r — rr =1, 
rotated about its conjugate axis, is 

a2 "^ o2 62 ^• 

The equation of the hyperboloid of two 

aj2 2^2 
nappes, generated by the hyperbola, -^ — i:; = 1, 

a2 62 

rotated about its transverse axis, is 



x2 



22 



62 62 



= 1. 



THE PARABOLOID. 

The equation of the paraboloid of revolu- 
tion generated by the parabola, x2 = 4 az, 
rotated about its axis, is 

a;2_|-2/2 = 4az. 



GENERAL EQUATION_OF CONIC 
SECTION. 

The general equation of any conic section, 
for which the F-axis coincides with the direc- 
trix and the X-axis passes through the foci 
normal to the directrix, is 

(x-A;)2+2/2 = e2a;2, 

where h is the distance from the directrix to 
the focus, and e is the eccentricity. 



DIFFERENTIAL 
CALCULUS. 



Variables will be represented by u, v, x, 2/, 

f and z, and constants by a, b, m, and n. < co 

D will be used as the sign for the deriva- g j 

tive, and d as the sign for the differential. S " 

Sin~i a: = angle whose sine is x. ^ " 



D (fx) = 



dx 



ddoge 


U) 


du 

=z , 

u 




diloga 


u) 


= loga< 


It 





y 


dx — x 


'dy 




2/2 





23 



a 



o 



.'. To obtain the derivative of any func- 
oion, drop the differential of the variable from 
the differential of the function. ^ | 

DMy)=I>y{fy)'D^y. 

da = 0. 

d (av) =a'dv. 2 ^ 

d (u + v-\-x) =du-\-dv-{-dx. 

d (x*y)=ydx-\-X'dy. o 

d(u'VX'y . . .) = (vx'y . . . ) du^ 
(u'X'y . . . )dv+(:u*vy . . . ) dx-{- 
(u'vx . . . ) dy+ ... ^ 



q: 

in 

o < 

ai S 



I "1 



h -t 



24 



where 



DIFFERENTIAL CALCULUS 



M = logae. 



d(jby)=^by'\ogab 

dxa = a'X^~^*dx, 



dy 

m' 



2Vx 
d (sin x) = cos x • dx. 
d (cos x) = — sin a; • dx. 
d (tan a;) = sec^ x • dx. 
d (cot x) — — cosec2 x • dx. 
d (sec x) = sec x • tan x • dx. 
d (cosec x) = — cosec x • cot x • dx. 
d (vers a;) = rf (1 — cos x) = +sin x • dx. 
d (covers x) = d (1 — sin x) = — cos x • dx. 
d (sin-i x)=dx/ Vi-x^. 
d (cos-i x) = -dx/ Vl-a;2. 
d(tan-ix)=dx/(l+a:2). 
d (cot-i x) = -dx/(l+x2). 
d (sec-i x) = dx/(x >/x2-l). 
d (vers-i x) = dx/ v^2 x — x2. 
d (covers~i x) = — dx / v^2 x — x^. 



jTo differentiate a function: 

1. Find the value of the increment of the 
function in terms of the increments of its 
variables. 

2. Consider the increments to be infinitesi- 
mals, and in all sums drop the infinitesimals 
of higher order than the first, and in the 



DIFFERENTIAL CALCULUS 



25 



remaining terms substitute differentials for 
increments. 

For the maximum value of a function the 
first derivative is zero, and the second deriva- 
tive is negative. 

For the minimum value of a function the 
first derivative is zero, and the second deriva- 
tive is positive. 

If -r- assumes the form - » then 
fx 

Px _ D jFx) 
fx'^Difx)' 

Taylor's theorem is 
f(x+h) = fx+h'D(fx) + ^'DHfx)+ . . . 

/a:=/(0+a;)=/(0) + 

X'D(fO)-\-^'DHfO)+ ' . . 



^■3 
33 



< CO 

o o 

I- z 

UI < 

a: I 

O o 

Ul ui 



The radius of curvature for a curve, y =fx, is 



da d^y 

where s is length of curve. 



{dsY 
dx-d^ 



2 s 

s u. 

o 






INTEGRAL CALCULUS, 



i dx = x 



+C, where C is the constant of 



integration. The constant C must be added 
to all of the following forms. 



Cidx+dy+dz . . . ) = 

Cdx-{- Cdy+ Cdz+ 

/-■ 

Jf =log,x. 

Ca'''dx=:r^ 
J log^a 

/ 

/ 
/ 
J 
/ 
/ 



n+i 

'dx = 
e'''dx = e*. 

dx = a'', 
smx*dx= — cos X or vers x, 
cos a; • dx = sin a; or — covers Xo 
sec2 X • dx = tan x. 
cosec^ x*dx= — cot x. 



sec x • tan X'dx = sec x. 



INTBGRAL CALCULUS 



27 



I cosec X • cot X' dx= — cosec x, 

I tan X'dx = log (sec x) . 

I cot a; •c?a; = log (sin x). 

I cosecx •c?a; = log (tan -)• 

I sec x • dx = log I tan (| + 7 ) • 

= _i.cot-(5y 

a \a) 

C dx 1 , lx-a\ 

r^:^==sin-.(2)=-cos-.(2). 



dx 



= l0g(x+ A/a;2±o2), 



J Va;2 _1- a2 

/rfx 1 , (x\ 

— = - • sec-1 (-), or 

= cosec~i (- j 

/I = vers-i ( - 1 , or 
V2 ax -x2 \«/ 

as — covers-i f^j 



J CO 
Ul o 

^^ 

- o 



< CO 

o o 

H z 

Ul < 

CE I 

o o 

UJ lU 



o < 

St 

LU 2 



-J 2 



/ 



h-i 



28 INTEGRAL CALCULUS 

Cf(x)dx = Fx+C, if 

d {Fx) =fx • dx, 
\ a * dx — a I dx. 

i X'dy = xy— | ydx. 
j ^F^ " P [a+hx-a . log (a+6x)]. 

/(^iSi = ^ t^^ ^^+^^)+ ^] • 

J ^+6^ = 6-3 1"^ 2a(a+6a:) 

+a2'log(a+6r) 1. 

/SS^ = 5i [«+6^-2o.log(a+6x) 

o+6a;J 

/ dx _ 1 . /£+^\ 
X {a-\-bx) a \ x J 

/ dx 1 _ JL 1 /ci-\-bx\ 

X (a+6a:)2 ~ ^^+6^) a2 ^ ^^ \ a; / 

/ dx _ _ ii_ _L A I / c^ + feA 
x2 (a+6x) aa; "*" o2 * °^ V x /* 

vben a>0 ^nd 6>0. 



h 



INTEGRAL CALCULUS 29 

dx 1 , Va+ajV^ 

log 



when a>0 and 6<0. 

/ dx ^ X 1 /* dx 

(a+6x2)2 2 a (a+6a;2) "^ 2a J a + bx^ 

/ dx _ _J_ x 

(a + 6x2)^+1 ~ 2na ' (a +6x2)^ 

. 2n-l r dx 
"*" 2na J (a+6a;2)^' 

Jx^'dx _x _a r _dx 
a+6a;2 ""6 b J o^ 



t+6a: 



/_a;2.da; 
(o 



+ fea;2)^+i 2n6(a + 6x2)** 



^2n& J (a+fea;2)' 
a: (a + 6x2) 2 o ^^ Va + 6x2/ * 

/ dx ^ 1 h r 
x2 (a+6x2) ax a J a 



+6x2 



/ rfx ^ 1 r <;x 
x2 (a+6x2)«+i a J x2 (a + 6x2)^ 

_6 r dx 

a J (a+6x2)«+i' 

rx"».(a+6x'')-^.dx = 

x"^-^-fi»(a+6x^)^+^ 
6(nP+m+l) 

6(nP+m+l) J^ (.a-^ox; ofa;. 



I- z 

lU <£ 

a I 

o o 

111 LU 



O < 



30 



INTEGRAL CALCULUS 



or 



nP+m + 1 
anP 



or 



or 



a (m+l) 
" a (m+l) J ^"^-^ • (a+fe^'^)^- rf^. 



an(P+l) 



nP-\-m-\-n-]-l 
an(P+l) 



ra;^.(a + 6x'*)-P+'.da:. 



/ 



dx 



ax2+fex+c 
2 



^ _ / 2a2;+5 \ 

V4 ac — 62 \ V4 ac — 62/ 

1 / 2ax+6- V62-4ac \ 

-4ac \2aa;+6+ V62_4ac/ 



/ 



V62 

a; 'cZx 



= J7^ -log (ax2 + 6a;+c) 
dx 



a.x2+6x+c 2 a 

2aJ ax2+6x+c 



I xv^ 



■\-hx»dx = 



2(2a-3 6x) (a+6x)i 



15 62 



Cx^-^ 



+ bx *dx = 



2 (8 gg- 12 a6x + 15 62x2) (a+6x)i ^ 
105 63 



INTEGRAL CALCULUS 

dx 2x''^a+hx 

^-1 • dx 



31 



/ x'^-dx ^ 2a;^V 
V^+fci (2/1- 






+ 6x <.i5 7i+l)6 

(2n+l)5j V^4-5a; 
rfa; 2(2a-6a:) Va+6^ 



r— 



a-\-hx 
dx 



log 



362 



when a > 0, 



when a < 0. 



x'^Va 



+bx 



(n — 1) ax^~^ 



(2n-S)b r dx 
{2n-2)aJ a;^iV^+6^' 

, r da- 
+a I — — 

*/ xVo- 



+6a; 



/^__^^_ = l.lo ( ^ ] 

xVa^ — x'2 ^ \a+ Va'^ — x^J 

/ dx 
r Va2-a:2 • dx = I V^23 



+ Va2 — x2> 
Va2-x2 



X2 






C-) 



O O 
tu lU 
? 2 



O < 

it 

LU S 

S u. 
O 



32 INTEGRAL CALCULUS 

§ (2 x2 - a2) Va2-x2 + 1^ sin-i f-^ 
o o \a/ 

/" \/a2_ a;2 



,dx= Va2-x2 



- /a+va2-x2\ 

_«.iog^ _ y^ 

/Va2-a:2 ^ - Va2 -a:2 . , /x\ 

/dx __ X 

(a2-a:2)i a2Va2-a;2 

r(a2-a;2)f.c?x = 

I (5a2-2a;2) V^3^+ | a* • sin-i (0 ■ 

/x2 • da; ^ re . /x\ 

(a2-x2)i V^H^ ^^^ W 




^ 

a;2V^ 



Va:2±a2 



a;2Vx2±a2 



/ rfx ^ Vx2-a2 1 _^x 



INTEGRAL CALCULUS 



33 



/ 



dx 



x3Vx2-fa2 



2 a2x2 "^ 2 a3 ^^ x 



/ 



Vx2±a2.rfa;: 



a2 



^ Vx2±a2zb -log(a;+Va;2±a2), 



rx2 Vx2±a2 .da;= | (2x2±a2) V^^'db^ 



a* 



-^log(a:+^a;2±a2). 
o 



/Va;2-a2 , ^^- a 
•cZx= Vx2 — a2 — acos-i - 
re X 



V'a;2d=a2 



6?x = Va;2+a2-a»log 



a+Va;2-|-o2 



/ 



a;2 



■da; = 



_ Va:2±a2 



+ l0g (X+ Va;2_{_o2). 



f J"'"^"^ _ = % Va:2±a2T ^' log (x+ ^^2 d=a2). 



Vx2±a2 2 "^ ^ ' 2 



(x2±a2)i a2Vx2±o2* 

a:2drc — 



/ 

J (x2±a2)i Vx2±a2 
r(x2±a2)ida:= | (2 x2 ± 5 a2) V^^±^ 



+log(a:+Va;2±a2), 



3 a* 



log(x+ Vx2±, 



o < 

uj S 
S u. 

o 



a « 



34 



/ 
/ 

/ 

/ 



INTEGRAL CALCULUS 

dx 



x'^dx 
V2arc-x2 



dx 



vers~i - 
2 a 



"*-iv2ax-a;2 



(2m-l)a r x^-i.d 
^ J V2^^ 



-2 



a;"*v2aa;-x2 
+ 



V2aa;-x2 
(2m-l)ax"* 



(2 m- 



1)« J 



da; 



2.m-iV2ax-x2 



V2aa;-a:2.dx= ^y^ V2aa:-x2 



, a2 . , a:— a 

+ -K sin-i -— - . 

J a 



Jx" 



V2aa:-x2.da;= - 
. (2m+l) 



a;™-i(2ax-x2)i 



m+2 



m+2 
- I x"^i • 'N/2aa;— x2 • dx. 



V 2 ax -x2 



/ V2a 
X 



dx = 



(2ax-x^)^ 



(2 w -3) ax"* 
m-3 r^2ax-x2 



(2m-3) 



-J- 



dx. 



s 



dx 



vW+fex+c 
1 



V^ 



log (2ax+6+2V^Vox2+6x+c). 



/Vax2+6x+c . dx = 1^^£±? Vax2+6x+c 
4a 

_ / b2-4ac \ r 

\ &a ) J ^c 



ax24-6x+c 



J^ 



f^ 



INTEGRAL CALCULUS 35 

dx 1 . . / 2ax — h 



—= sin-i (-— =) 



2 ax — 6 



ax24-6x+c«dx 



62+4 



4a 



V-ax2+6x+c 



/ 



xcix 



4ac C dx_ 

8« J V_ax2H-6x+c' 
V-i-ax2+6x+c 



V±ax2+6x+c 



zfca 



- r 

2a J 



c?x 



/x^ 



±ax2+6x+c»rfx = 



v±ax2+6x+c 

(=baa;2+6x+c)i 
3a 



=F — r>/-i-ax2+fex+c«dx. 



/x 1 
sin2 X • dx = - — J sin (2 x). 



2 4 



/x 1 
cos2 X • dx = - + - sin (2 x). 

I sin2 X • cos2 X • dx = - f X — - sin 4 x j 

J sec X • CSC X • dx = I -: 
J sinx»cosx 



= log tan X. 



/ 



sec2 X • csc2 X • dx 



/ dx 
sin2 X • c 



sin*™ X • cos'^ X • dx = 



cos2x 
= tan X — cot X. 

— sin^^i X • cos»+i X 






m- 



sin^"2 X • cos^ X • dxi 



f ^ 



o < 



;!i z 



ii 



36 INTEGRAL CALCULUS 

sin"^+i X • CDS'*"! X 



or 



m-\-n 



■\ r— I sin"* X • cos'^-s X • dx. 



m+n J 

fain^X'dx^ 

sin*^-i re • cos a; , m — 1 /* . ^_^ 

I sm"*-^ X • dx. 

m m J 

I Q,O^X'dx = 

sin x • COS**-! a; , n — 1 /*„_», 

/ sin"^ X , _ 
cos"* a; ~ 

l)cos^-ia; n—\ J cos 

/ 



(n — 1) cos^-i a; n—\ J coa^~^x 

cos'* a; J 
-7-— — • oa; = 
sin*" a; 

— cosH-ix m — n — 2 Cco^xdx 

(m-1) sin'^-ia; m-l J sin^'^a; " 

/cfa; _ —cos a; . m— 2 /* dx 

sin*" x ~ (m-l) sin"*-i a; m-l J sin"*^ x ' 

/dx __ sin a; i^~2 C ^^ 

cos'' X ~ (n — 1) cos^ia; n— 1 J cos*^-2 x ' 

ftan^ a; . rfx = *-^^^ - f tan^"^ a; • da;. 

cot"~^ a; /* 

cot'' x»dx= — — I cot^^-z X • dx. 

n-1 J 

/dx _ 
a+b cos X 

ifa2>62; 



^ 



INTEGRAL CALCULUS 37 



V6-ataii|+V5-|-a 
log 



ifa2<62. 

—x^ cos x+m j x^~^ cos x dx 
I x^ • cos x»dx = 

a;"* • sin X — m | x™~i • sin a; • dx 

/ sin a; ^ _ x^ a;^ _ a;^ 

x ''''"''" 313 "^51^ Til 

/sinx _ —1 sina; ■ 1 rcosxdx 
1^ m-l '^^'^ m-1 J x™-i 

/cos a; - , x2 x* a^ . 

-^dx=logx-2^ + j|^-^+ . . . 

/COS a; _ —1 cos a; 1 /'sin a; da; 

X™ ^ ~ w-1 * x"^-i m-1 J x"^i ' 

I X sm~i X'dx = 

i [(2 x2- 1) sin-i x+x Vi-x2]. 



x"^ sin-i X • dx = 

x'^+isin-ix 1 Tx'^+idx 



/ 

+1 sin-i X 1 r x^+^i 

n+1 n+1 J Vf^ 

/ 



x** cos~i X • dx = 



x^+icos-ix !_ r x^+idg 

»+l n+1 J VJ^' 



f ^ 



I t 



x^ Si 



38 INTEGRAL CALCULUS 

I x^ taii~i x»dx=- 

n+1 n+1 J H-x2 ' 

a a J 

-^ dx = • — — : -\ I •^— , ax, 

x^ n — \ x^-^ n — 1 J x^-^ 

e**^ log a; • da; = • ^ I • 

a a J X 

/nnr • / \ J „,/« slii [na?]— 71 COS [tixK 
e«« sm {nx) • da;=e** f — ^ITi — " ) * 

Je- cos (nx) dx=e- pcos (nx)+nsm(nx) J^ 

/ V/|^ • ^^= V(a+x) (6+x) 

+(a-6)log(V^^^+V6+^). 



= 2 cot-i i/~ 

V(a; -a) (6 -x) 



r ^ ^^ = 2 cot-i 1/^-=^ 

•/ v(a;-a) (6-x) ^"^ 



•2sm- 



• -1 i/^— « 
▼ o — a 

/- 

e/ X 



dx 1 , Va2+^-a 
, = — log ^ • 



/ 



w 



dx 2 , x2 

= — sec""^ 



X^T^^Q? ^^ <^ 



THEORETICAL 
MECHANICS. 



«, 



NOTATION. 

A = area. 

a = angular acceleration, 
a = linear acceleration. • 
an = normal acceleration. 
at = tangential acceleration. 
C = component of a force. 
F = force. 
F^= normal force. 
F< = tangential force. 
^ = acceleration due to gravity = 32.2. 
(The exact value is 32.1808 - 
0.0821 cos 2 L, where L is the 
latitude.) 
J^ = moment of inertia referred to center 

of gravity. 
Igg. = moment of inertia about an axis 
through the center of gravity and 
parallel to the X-axis. 
Jlf = moment of a force. 
m = mass = weight -v-g. , 
R = resultant of a system of forces. 
(S = space. 
V = velocity. 
vq = initial velocity. 
Wf = tangential velocity. 
y, z = rectangular coordinates of a point. 
Pi 6 = polar coordinates of a point. 
p = distance from pole to center of gravity. 
39 



< CO 

o o 
I- z 



m 

CO -i 

o < 



o < 

ai 5 



J 



40 THEORETICAL MECHANICS 

STATICS. 
Equilibrium of Forces. 

Concurrent Forces in Equilibrium in One Plane, 







Fig. 18. 



Non-concurrent Forces in Equilibrium in One 
Plane. 




Sikf = 0. 
XM=XC^y+i:CyX. 



Fig. 19. 



If three forces are in equilibrium they must 
be concurrent or parallel. 



If a system of non-concurrent forces in 
space is in equilibrium, the plane systems 
formed by projecting the given system upon 



THEORETICAL MECHANICS 



41 



three coordinate planes must each be in 
equilibrium. 

A couple consists of two equal and opposite 
parallel forces acting on a rigid body at a fixed 
distance apart. 

The moment of a couple is equal to the prod- 
uct of one force by the distance between the 
two forces. 

Centroid of Parallel Forces. 



/2 = SF. 

- _ SFx 

For a variable 
pressure, 




/ 



xFdx 



I 



Fdx 



Fig. 20. 



Center of Gravity of an Area. 

- _^Lxj_dA 

I i xdxdy 
I I dxdy 

XdA 

j I y dxdy 




y = 



Fig. 21 



// 



dx dy 



ai S 

S u. 

O 



42 THEORETICAL MECHANICS 

If 2/2-2/1=/^. 



dx 



j x{y2- 



y-d dx 




y: 



I (2/2-2/1) dx 

/- 



Fig. 22. 



/ 



>fx*dx 



fx 'dx 



Center of Gravity of a Mass. 
For a homogeneous mass, 

Z 




"Zxdm 



Fig. 23. 



I I \ xdxdydz 



f f {^^^y^^ 



= I 



22/ dm 



- _ S2 cZm 



///' 



<ia; dy dz 



US 



dx dy dz 



/// 



z dx dy dz 



/J7 



dx dy dz 



THEORETICAL MECHANICS 43 



Moment of Inertia of an Area. 



dx 



= j j y^dxdy. 
'^ = j j x'^dxdy. 



Fig. 24. 



♦— iCrH 



-X ► 




TT"^ 






= I yHx2-x^dy 
= j y^'fydy. 



Fig. 25. 



o < 



/o=Sp2dA 




O cj 



Fig. 26. 



44 THEORETICAL MECHANICS 

Moment of Inertia of a Mass. 
If A; is a constant, equal to the density 
divided by g. 




dx dy dz 

(a;2+2/2) dxdy dz. 



Rg. 27 
I^^p^dm 

-"Iff'' 

Product of Inertia of an Area. 

J = '2xydA= I I xydxdy. 

For the principal axes, J is zero. 
Radius of Gyration. 

Transformation Formtdse. 

I, = I,,-\-Ad^ 
or 

I^ = Ig^+md^, 

*^ xy^^ '^ c-g. "r A/c/l, 

where h, k are the 
X coordinates of the cen- 
ter of gravity referred 
to X-X and Y-Y. 




Fig. 28. 



THEORETICAL MECHANICS 



45 




+ Iy Sill2 d 

J'xy = Jxy^O&2d 



+l(/. 



7Jsm2a. 



To determine 
the value of B 
which will make 
X'-X' a principal 
axis, 

2Jxy 



tan 20 = 



I. 



Ellipsoid of Inertia. 

The moments of inertia about all axes 
through any given point of any rigid body 
are inversely proportional to the squares of 
the diameters which they intercept in an 
imaginary ellipsoid, whose center is the 
given point, and whose position depends 
upon the distribution of the mass and the 
location of the given point. This ellipsoid 
is the ellipsoid of inertia for the body. The 
axes which contain the principal diameters 
of the ellipsoid are called the principal axes 
of the body for the given point. 

Circle of Inertia.* 
For any plane figure, lay off OX parallel to 
Z-X, 

D 



OA = I. 




OB = I 



y 



BQ=^BA, 

circle through C with 
center at Q, 
* See Maurer's Technical Mechanics, Ap- 
pendix B, or Civil Engineers* Pocket Book. 



Fig. 30. 



46 



THEORETICAL MECHANICS 



Then 



CD parallel to X'-X' (Fig. 29), 
DE perpendicular to OX, 

QF = EQ. 

OE=r^, and OF =I\. 

ED = J' 



The principal axes for the given point are 
parallel to CM and CN. 

J is positive above and negative below OX, 

DYNAMICS. 
Velocity and Acceleration. 





ds 




dv dh 
^~ dt ~ df^' 




Uniformly Accelerated Motion. 


If 


a is constant, 




v = V(i-\-at. 




S = Voti-hat^ 




V^-Vq^ 



2a 



(vq+v) t. 



vdv = a 



Falling Bodies. 
For a body falling in a vacuum, a = g, hence 
v = Vo+gt. 

S = vot+^gt^ 



2g 



= 2 (^0+^) i' 



THEORETICAL MECHANICS 47 



Force and Acceleration. 

F = m, ' a= — 'U. 
9 



Direct Central Impact. 
For two inelastic bodies, let 
wii = mass of first body. 
7712 = niass of second body. 
Vi = original velocity of first body. 
V2 = original velocity of second body. 
V = common velocity after impact. 

Then v = ; • 

mi-\-m2 

For two elastic bodies having velocities 
ki and k^ after impact, 

miVi-\-'m2V2 = Tniki-\-m2k2. 

The product of mass by its velocity is 
momentum. 

The sum of the momenta before and after 
impact is constant. 




Virtual Velocities. 
F = force. 

F= direction of motion of P. 

du = virtual velocity of force. 

-rr = velocity of force. 
at 

-T- = velocity of P. 
at 

F 'du = virtual moment of force. 

The virtual moment of a force is equal to 
the algebraic sum of the virtual moments of 
its components. 

For a system of concurrent forces in 
equilibrium, 



O o 



H-" 



48 



THEORETICAL MECHANICS 



For any small displacement or motion of 
a rigid body in equilibriima under non-con- 
current forces in a plane, with all points of 
the body moving parallel to this plane, 

i:F'du = 0. 
Curvilinear Motion of a Point. 



Vt = 



ds 
dt 




Fig. 31. 



-©"+(i)'- 

_dv _dH 
'^'~ dt~ dp- 

=aa;Cos0-i-a„sin0. 



where r is the radius of curvature. 



^4 = m • tta; cos 0+m • a^ sin 9 = m • a<. 



2 



= j a^ ds. 



Projectiles. 
Neglecting the resistance of the air, 

x = Vq cos 6 • t. 



y = Vo3md't— -^gf^, 



or 

y = x tan 6 — 



gx' 



2 Vf? cos2 d 




Fig. 32. 



THEORETICAL MECHANICS 49 



Horizontal range, 



v^' 



Xr= — sin 2 6, 
9 

wiiich is a maximum for = 45°. 
The greatest height of ascent is 

Translation of a Rigid Body. 



R^= J a. 



dm. 



P^^ 



.dm 



vX 



Fig. 33. 



Fig. 34. 



The resultant force must act in a Une 
through the center of gravity and parallel to 
the direction of motion. 

Rotation of a Rigid Body. 

Let be the axis of rotation. 

= angular space passed over by any line 
from O. 

ct= angular accelera- 
tion. "*■ 



oj = angular 


velocity. 


Then 






03 = 


dd 
dt' 




a = 


do3 
dt ~ 


dW 
' dt^' 


<adco = 


■■add. 






Fig. 35. 



o a 

si 



h-j 



50 THEORETICAL MECHANICS 
For uniform acceleration, a = k, 

0) = Ci3Q-{-kt. 



e = (aot+-kt^ 



2a 

_ COQ + tO 

For a point p distant from O, 

2)< = p«C0. 

a=p *ot. 




For a mass m concentrated p distant 
from 0, 



THEORETICAL MECHANICS 51 



Precessional Rotation. 




Fig. 37, 



£0 = velocity about axis 
of spin {OP) in radians 
per sec. 

n = velocity about axis 
of precession {OZ) in 
radians per sec. 

7 = moment of inertia 
about axis of spin. 

r = torque ( = TFp for 
equilibrium) . 

!r=coQ/8in 9, 

For 9 = 90°, T=o3^1. 



Center of Percussion or Oscillation. 

If an unsupported bar upon being struck 
at a begins to rotate about 6, then a is the 
center of percussion for 6 as a center, and h 
is the center of instantaneous rotation. 



Fh = 



/' 



,2. 



dm 



dF = a *p *dm. 



="/"• 



dm 



pm p 



(i 



Fig. 38. 



o a 



d) 00 



52 



THEORETICAL MECHANICS 




Pendulum. 

r = tiine of oscillation 
from one extreme position 
to the other. 

r = radius of gyration. 
P'9 



Work, Energy, and Power. 

Work (w) is equal to force (F) multiplied by 
the distance (S) through which it acts. 



Power (L) is the rate of doing work. L ■■ 



w 



Energy is the capacity to do work. 



The energy of a moving body, K.E. = -miP. 



The kinetic energy of rotation '\^K.E. — -zI'ufi, 



Friction. 

F = friction. 
i\r = normal force. 
/= coefficient of friction. 
F=f'N. 



m 



N 

Fig. 40. 



Angle of friction, 0=tan-i -^ 



THEORETICAL MECHANICS 53 

Average values of / for motion are as 
follows: 

Wood on wood 25- . 50 

Metal on wood 50- . 60 

Leather on metal . 56 

Leather on metal, lubricated .... 0.15 

Metal on metal, — dry 0.15-.24 

Lubricated surfaces: 

Ordinary 0.08 

Best 0.03-0.036 

For values of / for rest add 40 per cent to 
above values. 



Friction of Belt 


• 






\ ''J 




F 


t 


V 


Fig. 41. 


Fig. 42. 


dF=f'Nds=f 


r 





ds = rdd, and f*dd=^' 
or Fi-e^'^i = Fz, where Oi is in radians. 



o < 
< ^ 



a CO 

f3 



MECHANICS OF 
MATERIALS. 



Si 

o 



0-2 1 

fl c 

^^6 






•5lOOqg 



•j^A 



•i^p^a^g 



.fl 



ICQ 



I— I 



S s f^' 
o £ o 



oooooo< 



© o 



0000 

8888 






I O O oooooooooooooooo 
I o O looooocooooooeooo 

»0 O WOWO^lC^N-^OnClMW*© 



S s f3 






a d 
®.2 



0000000 ©ooooooooooooooo 
O O O O O O O O0OOO00COU50O00OO 
O O O O 10 O ^O o«o«ot-ot-owio«ioiooo 
" 10 ■* w «c« 



■cD<aDooi>.«o«OTt<oo 



000000000 
000000000 

CO O O O O O O O o__ 
t>r t>r 05 00 I>^ 00 t>^ O 



j8d sSuiy; 






00 00»00 0»0l:^0»0»00»000 
iOOOOSOi(MeOlOCO<MTt<COCOCO<M-<^ 



^^ ^51 ^J1 TjH ^H l^H ^H 






3 GO; 



1^ n :3 o O 









■®co 


•|i 


t§ 


u, rt 


> 00 


d U 


•^3 


bC^ 


i§ 


-^ C 


ra"« 


sa 


§^ 


T3>^ 


§1 


fl ti 


•s^ 


^nfl 


©.2 


■^ S 


-tj g 


b 


-M 0. 


3S 


1^ « 



c3 O 



54 



MECHANICS OF MATERIALS 55 



NOTATION. 

A — area. 

h = breadth. 

d = depth. 

E = modulus of elasticity. 

e = total deformation. 
F = force. 

I = moment of inertia. 
Iq = polar moment of inertia. 
J = product of inertia. 

I = length. 
M = moment. 
R = resultant of forces. 

r = radius of gyration. 
5 = unit stress. 

s = section modulus. 
y = vertical shear. 
W = total weight. 
w = weight per lineal unit. 
A = maximum deflection. 

€ = unit deformation. 

Direct Stress. 

For an axial tensile or compressive force, or 
for simple shear, 



I , e eA 

\ For tension or compression the deformation 
'i 'S measured along the axis of the member, and 
' *or shear it is measured at right angles to the 
1 ixis of the member. 



o < 

z oc 

S u. 
O 



56 MECHANICS OF MATERIALS 

Eccentric Loads.* 
F 




Fig. 43. 

Consider a section a-a perpendicular to 
axis of a bar, and take axes of coordinates 
through center of gravity. 

Let re, 2/ = coordinates of any point of section. 

n-n = neutral axis. 

D = distance of any point from line through 
center of gravity and parallel to neutral axis, 
positive toward P. 

i>o = value of v for neutral axis. 

F = force or resultant of forces acting at P. 

iV = component of F normal to section 
considered. 

^0 = unit stress at center of gravity. 
'^0- -7' 



* The method here presented is taken from 
a paper by L. J. Johnson, M. Am. Soc. C. E., 
*' An Analysis of General Flexure in a Straight 
Bar of Uniform Cross Section," Trans. Am 
Soc. C. E., volume LVI, p. 169, 1906. 



MECHANICS OF MATERIALS 57 



o 

= aSo (y cosa — X'Sina) 

jy N -Xpiy — x tan a) 
A J — Iy tan a 



jy N -ypiy — x tan a) 

A Ix~J tan a 

j^ N * pp{y — x- tan a) cos 9 

A J — ly o tan a. 

jy N ' pp(y — x * tan a) sin 9 

A I^ — J* tan a 

^ N {yply - XpJ) y+N {xp[* - ypJ) x 

^A^ IJy-J^ 

N 
= ~+N>PpX 

[ {ly smd — J' COS 6) y-\- (Iz cos — J sin 6) x l 
IJy-J' J* 

N 
In the above equations -r is the portion of 
A 

S which is direct stress, and the other term 

is the portion due to the bending moment, 

M = N'pp. If s represent the section 

modulus 

/ Izly-J^ \ 

\(Iy sin 6 — J ' cos d) y-]- {I^ cos d — J-smd)x/ 
then 

A s 



Note. — The values of the section modu- 
lus given in the handbooks are computed 

/ 



from the formula 



, which is the value of 






58 



MECHANICS OF MATERIALS 



s for J — and for P located on Y-Y. 
angles and Z-bars J does not equal zero. 
In the above equations, 



I 



For 



tana = 



/. 


-J . 


tan 9 




J 


-h' 


tan^ 




h 


COt0 


-J 




J cotd- 


-h 




h 


COS 6 


— J • sin 


d 



J cos Q — Iy sin 6 

For any bar having a section which is 
symmetrical about either axis, ^ = 0, and 
the values of *S become 



N 



Q -■ , XT ( ly smd ' y^I^cos e ' x\ 

'S=^+^-pp(^ jj y 



If for a symmetrical section, P is on Y-Y, 
then sin = 1 and cos = 0, or 



N N-pp-y 



.V M-y 
A^ I^ ' 



->-N- 



.P4---l-l-ld 



Fig. 44. 



For a rectangular section, for which N is 
applied on Y-Y and p distant from the axis 
of the bar, the extreme fiber stresses are 



-S = 



!(-«i) 



MECHANICS OF MATERIALS 



59 



EQUATION OF NEUTRAL AXIS. 

The equation of the neutral axis for an 
eccentric load is 



KERNEL OR CORE-SECTION. 

The kernel of a section (sometimes called 
the core-section) is the area within which P, 
the point of application of the resultant of the 
forces, must fall in order that the stress shall 
be of the same sign throughout the section. 
It is the area bounded by the locus of the P'3 
corresponding to a series of neutral axes 
touching the periphery of the section but never 
crossing the section. For every side of the 
section there will be an apex of the kernel. 
If x^, TJa and x^, yi^ are the coordinates of a and 
6, which are two consecutive vertices of the 
section, then the coordinates, a;„j, 2/^5, of the 
vertex of the kernel corresponding to the side, 
ah, of the section will be 



^ah= - 



yab=- 



(x„ -xt) J- (Va - Vb) ly 

^ i^aVb-^byal 
ix^-Xi) Ix-iVa-Vb) J 



A {x^Vh-Xky^ 
If ah is parallel to X-X, then 



^ab= — 



yab=- 



h 



If ah is parallel to F-F, then 

ly J 



^ab= - 



A-x, 



A'X^ 






h-i 



60 MECHANICS OF MATERIALS 

The radii vectores of the kernel are lengths 
which for any d need only be multiplied by 
the area of the section (A) to give the sec- 
tion modulus 

/ Ixly-J^ 

\(/„sin e — j'coad) y-\-(Iy» cosd — J 'sin 



e)x) 



but these lengths must be considered posi- 
tive if measured on the opposite side of G 
from P. 

SECTION MODULUS POLYGONS. 

N M 

In the equation S= --r-\ (see Eccentric 

A s 

Loads), s is the section modulus. The sec- 
tion modulus polygon is the polygon the 
lengths of whose radii vectores are the 
graphical representations of the values of s 
for extreme fibers for successive values of 
6 from to 360 degrees. The section modu- 
lus polygon is a figure whose sides are parallel 
to the sides of the kernel of the given section 
but which lie on opposite sides of the center 
of gravity from the sides of the kernel. 
The most general value of s is 

Ixly-J^ 

{ly sind — J cos d) 2/+ (ly cosd—J • sin 6) x 

For any section which is symmetrical 
about either axis, s becomes 

Ixly 



ly sin d -y+Ix cosd-x 

For any symmetrical section for which P 
lies on F-F, = 90°, hence 

y 



MECHANICS OF MATERIALS 61 



If for any symmetrical section P lies on 
X-X, 6 = 0°, hence 

,= :^. 

X 

There will be one vertex of the s-polygon 
for each side of the polygon bounding the 
section. If x^, Va and Xf^, y^, are the coordi- 
nates of a and b, two consecutive vertices of 
the bounding polygon of the section, then 
the coordinates of the vertex of the s-polygon 
corresponding to the side ab of the bounding 
polygon will be 



^ab = 



(Xa — Xb)J- {Va - Vb) h 



yab= 



^aVb-^bVa 

(Xg - Xi) la- - (yg — y^) J 

^aVb-^bVa 

If ab is parallel to X-X, 



J 

If ab is parallel to F-F, 



2/a6 = 



^ab- 



ly 

— 1 

x„. 



Va 



Vab^-r^ 



For sections symmetrical about 
X-X, or Y-Y, J = 0, and the values of 



and — can 

Xg 



be found in the 



either 

Va 

handbooks 



issued by the steel companies, under the 
column marked "Section Modulus." The 
vertices can then be plotted and connected 
by straight lines to form the s-polygon. 
From this s-polygon the values of s for any 
) value of d can be obtained graphically. 

The most advantageous plane of loading for 
any section will be that having the greatest 
value of s. 



<i 



p-- 



62 MECHANICS OF MATERIALS 

DIAGONAL STRESSES, 
n 



F-^ 



—r 

/ 

f 
■7— 



L^. 



>F 



/ 
/ 

n 

Fig. 45. 

F = axial load. 

A = area of section normal to axis of bar. 
n-n = any diagonal section. 

d = angle which n-^ makes with axis. 
*S = umt axial stress. 
(Sa = unit shear along plane normal to axis. 

Sn = unit tension or compression normal to 
section n-n. 

Sgn = unit shear along section n-n. 
For combined direct stress and vertical shears 
S„=|(l-cos2^)+Sa.sin20. 



^8^=-. sin 2 0+5,. cos 2^. 



occurs when cot 2 0= — r-^ , and is 

max.S,= is±(s/ + |-) • 
The maximum value of Ssn occurs when 
tan 2 0=4- TT-o"* ^^^ is 

J Og 

max. 5,^= (5,2 -f I) . 



MECHANICS OF MATERIALS 



63 



For axial load only, Sg = 0, hence 

Sn=^(l-co32d) = S-am^d= -' sinz d. 

£i A. 

The mayiTTium value of S^ occurs when 
Q = 90°, and is then the unit axial stress. 

The maximum value of aS^^ occurs when 

S F 
= 45°, and is - or ^r-r' 

Zi Ji A. 



R,< 



THIN PIPES, CYLINDERS, AND SPHERES. 

<S = unit stress in metal. 

< = thickness of metal. 

d = diameter. 

p = unit pressure of liq- 
uid or gas. 

= angle which the di- 
rection of p makes 
with X-X. 




Fig. 46. 



For the transverse stress across a longi- 
tudinal section of a pipe or cylinder, 

Ri = R2= K^V' cos d= -p'd. 
Ri p'd 



For the longitudinal stress across a trans- 
verse section of a pipe, or for the stress in a 
thin hollow sphere. 



S = 



. 1 



rd't 



p • d 

T7 



which is one-half of the unit transverse stress 
in a pipe having the same diameter and thick- 






H-^ 



64 MECHANICS OF MATERIALS 



RIVETED JOINTS. 







Fig. 47. 

distance center to center of two consecu- 
tive rivets in one row. 

diameter of rivet or rivet hole. 

stress in unriveted plate in length a, 

thickness of plate. 

unit tensile stress. 

unit compressive or bearing stress. 

unit shearing stress. 

efficiency of joint for tension. 

efficiency of joint for compression. 

efficiency of joint for shear. 

number of shearing sections of rivets in 
distance a. (Notice that for butt 
joints each rivet has two shearing 
areas.) 
n = number of bearing areas of rivets in dis- 
tance a. 

F = tia-d) St=m'-rTrd^'Ss = n't'd'Sc. 



a = 

d = 
F = 

t = 

St = 

m = 





a-d 


e« = 


a 




m'TT- d^Ss 


Ca = 


4 . atSt 




n-dSc 


ec = 


aSf 



MECHANICS OF MATERIALS 65 

For maximum eflBciency, make et = eg = eci 
for which 

a = 



and 



m 'TT ' Sg 

4:nS, t 
mirS 



;(-'^l)- 



The allowable value of Sc is usually 2 Sg. 

For single riveted lap joints the maximum 
eflSciency is approximately 55 per cent, for 
double riveted lap joints approximately 70 
per cent, for triple riveted lap joints approxi- 
mately 75 per cent, and for triple and double 
riveted butt joints approximately 80 per cent. 



BEAMS. 

Vertical Shear. The vertical shear at any 
section of a horizontal beam is equal to the 
sum of the vertical components of the reactions 
to the left of the section minus the sum of the 
vertical components of the loads to the left 
of the section. 

For any beam the vertical shear upon the 
right side of the left support of any span is 

where • 

ikfi = the moment at the left support, 

iW2 = the moment at the right support, i- 

< 

w = the uniform load per lineal imit, ^ 

F = any concentrated load, 

a = the distance from the left support to F. 

I = the length of span. 



^ a 






66 MECHANICS OF MATERIALS 

Shearing Stresses. If y = vertical shear at 
any section, 

where Sg is the average unit shear. 

The actual unit vertical shear at any point 
is equal to the unit horizontal shear at 
that point, and may be determined by the 
following equation: 

where b is the breadth of the section at the 
given point, y is the distance of the point 
considered from, the neutral axis, and c is 
the distance from the neutral axis to the 
extreme fiber on the same side as the point 
considered. 

The maximum value of S^ occurs at the 
neutral axis, and is 

V r'' V 

max. Ss= j-^ \ y'dA= j—^'AiVi, 

where Ai is the area of the portion of the 
section on one side of the neutral axis, and 
2/1 is the distance from the neutral axis to 
the center of gravity of the portion of the 
section on one side of the neutral axis. 

For a rectangular section, the maximum 
unit shear is § of the mean unit shear. 

For Diagonal Shear, see Diagonal Stresses, 
page 62. 

Bending Moment. The bending moment at 
any point for any beam is 

M = Mi+ViX-hwx^-'SF(x-a), 



* See "Merriman's Mechanics of Materi 
als," page 269. 



MECHANICS OF MATERIALS 67 

where 

Ml = bending moment at the left support, 
1^1 = vertical shear upon the right side of 

the left support, 
F = any concentrated load upon the left 

of the section considered, 
X = distance from the left support to the 
section considered. 

For any beam of one span Fi is equal to the 
vertical component of the left reaction. 

The maximum positive moments occur at 

those sections for which -^— becomes equal to 
dx 

or passes through zero, that is where the shear 

becomes or passes through zero. The negative 

moments at the supports may have the largest 

numerical values, and for these points —, — does 

ax 

not equal zero, since the tangents to the mo- 
ment curve are not horizontal at these points. 
Theorem of Three Moments. For any two 
consecutive spans of a continuous beam, let 

Ml = moment at the left support, 
iV/2 = inoment at the middle support, 
^3 = moment at the right support, 
Zi = length of the first span, 
Z2 = length of the second span, 
Z = length of span for equal spans, 
ti?i = uniform load per lineal unit on first 

span, 
W2 = uniform load per lineal unit on second 

span, 
Fi = any concentrated load on the first 

span, 
i?2 = any concentrated load on the second 

span, 
ai = distance from first support to Fi, 
a2 = distance from middle support to F2. 






68 MECHANICS OF MATERIALS 

Then, for uniform loads only, 

Mih+2 M2 Qi+l2)+Mzk= - \ t^iZi^- i ^^2^23. 

For equal spans with equal uniform loads^ 

Mi+4.M2+Mi=-^wlK 

For concentrated loads only, 

Mih+2Mi{h+h)+Mzh 
= -F,(a,h- ^) - ^2(2 a,h-Sa2^+ ^) • 

Flexural Stresses. The tensile and com- 
pressive stresses in a beam, produced by bending, 

N 
can be determined by placing -j = in the 

formula for S given under Eccentric Loads, 
which gives 



For combined flexure and direct stress, the 
tensile and compressive stresses can be com- 
puted for prisms by the formulae given under 
Eccentric Loads, and for long members by the 
formulae given for Eccentrically Loaded Columns, 

Elastic Curves. The curve which is as- 
sumed by the neutral surface of a beam under 
load is called the elastic curve. 

The radius of curvature of the elastic curve 
is 

■g/^ dl^ ^dx^ 
~M dx'd^y d^y* 

from which the equation of the elastic curve 

can be obtained, for any particular case, by 

d^y 
placing M equal to EI — , and by making 

two integrations to obtain an equation in 
terms of x and y. 



MECHANICS OF MATERIALS 69 



The deflection of a beam at any given point 
is obtained by substituting the particular value 
of x in the equation of the elastic curve and 
solving for y. The maximum deflection occurs 
at the section for which 



dx 



= 0. 



(For particular cases, see Table of Beams.) 



TABLE OF BEAMS. 

Note. — The equations for elastic curves 
and the values of A apply only to beams of 
uniform section. 



Beams Supported at Both Ends and Uniformly 
Loaded. 



1 , w 


WMmmmwA 






V 7 ^ 


k 1 


V = Ri-wx. 
M=RiX-^wx^ 

= -wlx-^wx^ 

= lwx-\wx\ 


1 


1 Shear ^-^-^^^ 


Moment 

Fig. 48. 





2AEIy = w {-x^^-2lx^-Px), 



2/ = A when 2; = ^ * 

A- A w^*^_5_ ^^ 
384 £;/~384 EI 



o a 

III V 



a !» 



70 MECHANICS OF MATERIALS 

Beam Supported at Both Ends and Loaded 
with a Concentrated Load at Center of Span. 



Ri = R2=-^F. 



15 



Ro" 



Shear 



V = Ri, or V = R2. 
M = RiX, on the left of F, 

= Rix-f(^x-^, on 
the right of F, 



M, 



Moment 
Fig. 49. 



Elf^^^Fx, on the left 
ax^ 2t 

oiF. 
^SEIy = F{4:X^-3Px), on the left of F. 

48 El' 

(For both uniform and concentrated loads, 
combine the results for each.) 

Beam Supported at Both Ends and Loaded with 
a Concentrated Load Distant a from the Left 
Support. 

R2 = F-R, = F{jy 

V = Ri, on the left of F, 

= Rz, on the right of F, 
M = RiX, on the left of F, 

= Rix — F (x — a), on 
the right of F. 



---i' 



Tr7 



R# 







Shear 









Af. 



=-(-f) 



Moment 
Fig. 50. 



d^y 



EI ^, = Rix, on the left of F, 
ax^ 

= RiX — F (x — a), on the right of F. 



MECHANICS OF MATERIALS 71 

For the curve on the left of F, 

GEIy^F (l-^) x^-F(2al-3a^+j\ x. 

The maximum deflection (A) occurs at the 
section for which 



Vl2al-a^ 



--^^ ^=M-^)\^-'^- 



Beam Supported at Both Ends and Loaded 
with Several Concentrated Loads. 



R,= 



^F(l-a) 
I 



1X2 — — 7 — = S /" — iVJ. 



v=r^-^"f, 



M=R^x-^''F{x-a), 



The maximum moment (M"^) occurs at 
the section for which Ri—yY'F equals or 



passes through zero. 

For a system of movable loads the maxi- 
mum moment will occur under one of the 
loads, the loads being in such a position that 
the center of the span is midway between the 
center of gravity of all the loads and the section 
at which the maximum moment occurs. 

The maximum deflection of a beam loaded 
with several loads is the sum of the deflec- 
tions produced by each load at the section 



X ^ 






72 MECHANICS OF MATERIALS 

at which the maxunum deflection for the 
entire system of loads occurs. The deflec- 
tions produced by each load can be obtained 




Moment 



Fig. 51 



by means of the equation of the elastic curve 
for a single load. 

Cantilever Beam with Uniform Load. 

2^2 = 0. 

; V = Ri-wx. 

or if X is taken from 
the free end, 




lloment 
Fig. 52. 



M = - wx^. 



MECHANICS OF MATERIALS 73 

24 Ely = wx^-4: wlx^ + 6 wPx"^, 

8 EI 8 EI ' 

Cantilever Beam with Concentrated Load at 
the Free End. 




1 


1 
I 


Mm 


....^ 1 




Moment 




Fig. 53. 


Ri 


= F. 


R, 


= 0. 


V 


= Ri. 


M 


-Fil-x). 


M^- 


= Fl 




= F(l-x). 


6 Ely: 


= ^Flx^-Fx\ 


A- 


IFl^ 
3 El' 



Beam Fixed at Both Ends and Uniformly 
Loaded. 

Ri = R2=lwl=^W, 
V=Ri-wx, 






74 MECHANICS OF MATERIALS 



ilf = — — wl^-\- - wlx— 5 wx^. 



EI% = M,+ \wlx-\w^K 



i 



K — 



R2t 



Shear 



^^ 



M, 



^J^ 



Moment 
Fig. 54. 



^'^- 



By placing -r- = when a; = and when x=l, 
ax 

Mi = -^wP=-j^Wl=M^. 

2^EIy = w (-Px^+2lx^-x^), 

SS4:EI 384 '^** 



Beam 


Fixed 


at 


Both 


Ends and 


Loaded at 


the 


Center 


of 


the 


Span with 


a Concen- 


trated Load 












Ri = R2 


1 
2 


F. 








V = Ru 


on 


the left of F, 






^R2, 


on 


the right of F, 





I 



MECHANICS OF MATERIALS 75 



i FZ+ ^ Fx, on the left of F, 



on the right of F. 



Shear 



Ml'' 

A 1 r 



Moment 



Fig. 55. 



^fT^^ 



M, 



EI-^^ = Mi+ TT Fx, on the left of F, 
ax^ J 



= Mi+ -Fx- 



-(^-y 



on the right of F. 

By placing -j- =0 when x = Q and when x = -x' 
ax Z 

48 Ely = 4 Fx3 - 3 Flx"^, on the left of F, 

1 Fl^ 
192 Ej' 



III V 



76 MECHANICS OF MATERIALS 

Beam Fixed at Both Ends and Loaded with 
a Concentrated Load Distant a from the 
Left Support. 




7= El, on the left of F, 

= /22i on the right of F. 
Jkf = iWi+jKia:, on the left of F, 

= ilfi+Z2ix-F (x-a), on the right of ^ 

g;/ ^ =Mi+Rix, on the left of F. 
EIy = S Mix'^+Rix^, on the left of F. 



MECHANICS OF MATERIALS 77 



The maximum deflection (A) occurs at the 
2al 



section for which x = 



1+2 a 
2 MraH^ , 4 RxaH^ 



+ 



Continuous Beam with Uniform Loads. 

Wi = load per lineal unit on Zi. 

W2 = load per lineal unit on l^, etc. 
Tri = total load on Zi. 
W2 = total load on Z2, etc. 



^^^^^^^^^^^^^^^^ ^g_^ 




^^^^--^Nl^^^Ni;^'^^!^' 



Fig. 57. 

For a continuous beam supported hut not fixed 
at the ends, use the theorem of three moments, 
writing the equation for the first and second 
spans, for the second and third spans, and so 
on, to the end. Solve the simultaneous equa- 



78 MECHANICS OF MATERIALS 

tions, thus obtained for the moments at the 
supports. Then 

Tr -^2,1, 

T- Mz-Mi , 1 , 

V2b= 1 1-2 ^2^2. 

Vsa=W2-Vzb,etc. 

For equal spans with equal uniform load 
over the entire beam, the ends of the beam 
resting upon supports, the moment at any 
support is Kwl^ or KWl, and the vertical 
shear is Nwl or NW, where K and N have the 
values given in the following table. For 
many practical calculations the moment at a 
support one span from the end is assumed to 

be — — Wl, and for intermediate supports 

-Wl, 

12 

For a continuous beam with fixed ends con- 
sider an imaginary span to be added at each 
end of the beam, with the free ends resting 
upon supports. Then write the equation of 
three moments for each two consecutive 
spans, making Z = for the first and last spans, 
and compute the moments at the supports as 
shown above. 



Continuous Beam with Concentrated Loads. 

Determine the moments at the supports in 
a similar manner to that employed for con- 
tinuous beam with uniform load, employing 
the equation of three moments for concen- 
trated loads. 



MECHANICS OF MATERIALS 79 



u 

a 

o 

to 


^ 




', 


'. O 


.M 




', 


* H« 


^*l 


: : ': =" 55 


u* 




C' 




© 


»4» on 






« 


Ida e«0 


^ 


'. © 


•S 


2g se 


t!' 


I 9*0 


«S 


2S SS 


, ^ 


© >t*a 


-s 


ic(» on 


^' 


iHfc^ «*0 


«^ 




u- 


Hw «» 


^ 




u- 


© © 


© 


© © 


43 

S 

o 


^° 


1 1 




I © 


^" 


\ \ 




<=> < 


^' 


: : 


© 


^ < 


^" 


: o 


-« 


HS «S 


^" 


© -w 


HS 


«g ^ 


^" 


o © 


© 


© © 


No. 

of 

Spans 


^ (N 


CO 


Tt» lO 



80 MECHANICS OF MATERIALS 



STRUTS AND COLUMNS. 
.SziZer's Formula, 




Fig. 58. 



El 



dx2 



-Fy. 



-(*f)(^) 



Since y = a when x = 

F=EI 



or 



1=-©^ 



I I 

2' 2 
7r2 

Z2* 



^ must equal 



for round ends. 



For one end round and the other end fixed^ 
4 
replace I by -I and tt by 2 tt, which gives 

o 



A 4 



(9^ 



3 



For both ends fixed, replace I hy — I and tt 

by 3 TT, in the formula for round ends, which 
gives 



F=4LEI-. 

F 

4 = 4x2^ 



(9' 



MECHANICS OF MATERIALS 81 



Rankine's Formula. 
Gordon's Formula.) 



{Sometimes called 




Fig. 59. 
From the formula for eccentric loads for 
a symmetrical section (page 57), the maxi- 
mum stress will be 

F My 

where y is the distance from the neutral axis 

to the extreme fiber. 

p. 
But, / = Ar2, M = Fa and a = K—, where K i3 

y 

some constant depending upon character and 
condition of the column. Hence 



F S 

A 



m 



or 



The following values of K are recommended 
in the Civil Engineers' Pocket Book:* 



Material. 


Both 
Ends 
Fixed. 


One End 
Fixed, 

One End 
Round. 


Both 

Ends 

Round. 


Timber .... 
Cast Iron . . . 
Wrought Iron . 
Steel 


1/3000 
1/5000 
1/36000 
1/25000 


1.95/3000 
1.95/5000 
1.95/36000 
1.95/25000 


3/3000 
4/5000 
4/36000 
4/25000 



Hitter's Formula. Ritter's formula is the 
same as Rankine's formula except that the 

* American Civil Engineers* Pocket Book, 
p. 307. 



o a 



82 



MECHANICS OF MATERIALS 



expression -^ is used for K, in which Sg is 

' n is 



and 



the elastic limit of the material, 

9 
equal to tt^ for round ends, - ""^ for one end 

round and one end fixed, and 4 tt^ for both 
ends fixed. 

The Straight Line Formula. The straight 
line formula is 

A r 

where (7 is a constant depending upon the 
character and condition of the column. 

Merriman gives the value of C in the above 
equation to be 



= i^(: 



s^i 



.SnEJ 

which is obtained by making the straight line 
a tangent to the curve for Euler's formula 

0, the values 



I 



passing through the point S for - 

r 

of n being those given for Ritter's formula. 

Values of constants for the straight line 
formula, as determined by T. H. Johnson, for 
rupture, are given in the Civil Engineers' 
Pocket Book* as follows: 



Kind of Column. 


S. 


C. 


Limit 

l/r. 


Wrought Iron: 








Flat Ends .... 


42,000 


128 


218 


Hinged Ends . . 


42,000 


157 


178 


Round Ends . . 


42,000 


203 


138 


Structural Steel: 








Flat Ends .... 


52,500 


179 


195 


Hinged Ends . . 


52,500 


220 


159 


Round Ends . . 


52,500 


284 


123 


Cast Iron: 








Flat Ends .... 


80,000 


438 


122 


Hinged Ends . . 


80,000 


537 


99 


Round Ends . . 


80,000 


693 


77 


Oak, Flat Ends . . 


5,400 


28 


128 



* American Civil Engineers' Pocket Book, 
p. 308. 



MECHANICS OF MATERIALS 



83 



Some of the values of constants commonly 
used for designing steel columns, by the straight 
line formula are as follows : 



Member. 


S. 


C. 


Limit 

l/r. 


Author- 
ity. 


R. R. Bridges: 










Chords, L. L. . . 


10,000 


45 


100 


Cooper 


Chords, D. L. . . 


20,000 


90 


100 


Cooper 


Posts (Thru), L.L. 


8,500 


45 


100 


Cooper 


Posts (Thru),D. L. 


17,000 


90 


100 


Cooper 


Posts (Deck), L.L. 


9,000 


40 


100 


Cooper 


Posts f Deck), D.L. 


18,000 


80 


100 


Cooper 


Laterals (Wind) . 


13,000 


60 


120 


Cooper 
Am. Ry. 


Any- Member . . 


16,000t 


70 


*[iE8- 


Eng. and 
M. of W. 


Highway Bridges: 








Assoc. 


Struts 


16,000 


70 


*[\ll 


Ketchum 


Chords, L. L. . . 


12,000 


55 


*\u 


Ketchum 


Chords, D. L. . . 


24,000 


110 


* 125 
150 


Ketchum 


Posts (Thru), L.L. 


10,000 


45 


* 125 

Lloo 


Ketchum 


Posts (Thru), D.L. 


20,000 


90 


*fl25 
Ll50 


Ketchum 


Posts (Deck), L.L. 


11,000 


40 


niig 


Ketchum 


Posts (Deck), D.L. 


22,000 


80 


*{\n 


Ketchum 


Laterals, Wind . . 


13,000 


60 


*ri25 

lloO 


Ketchum 


Girder Stiff eners. 


12,000 


55 




Ketchum 


Buildings: 










Columns .... 


16,000 


70 


*[\l% 


Ketchum 


Columns .... 


16,000 


70 


♦fl20 
1150 


Chicago 



For cast iron columns, for which - does not 

r 

j exceed 70, the Chicago ordinance allows 
I 10,000-60-. 



For timber columns, the formula is changed 



to 



-(--1) 



* Main members and laterals, respectively, 
t Impact of live loads to be taken into account 

by adding 7 = 5 ^ , _„„ , in which aS = actual live 

JL/-t-oUU 

load, and L = length of bridge loaded. 



Si 



84 MECHANICS OF MATERIALS 

in which S is the allowable compressive stress 
along the grain, and d is the diameter. The 
Chicago ordinance (Mr. Benj. E. Winslow's 

formula) uses ^pr for C, for values of — not 

greater than 30. Ketchum's Specifications for 

Steel Frame Buildings gives C = -— : • 

Eccentrically Loaded Columns. By adding 

My 
the bending stress -j- to Rankine's formula, 

replacing M by Fe, and / by Ar^, the formula 
becomes 



b^4 



+• 



in which the constants are those given for 
Rankine's formula, e is the eccentricity, and 
y is the distance from the neutral axis to the 
extreme fiber, 

A more general formula for combining direct 
and bending stresses is 

F My ^ 

in which M is the apparent bending moment, 
y is the distance to the extreme fiber, / is the 
moment of inertia, E is the modulus of elas- 
ticity, and a and /8 are constants, /8/a being 
9.6 for a simple beam uniformly loaded and 12 
for a simple beam with a load at the center. 

The following formula for steel struts, given 
in Ketchum's Specifications for Steel Frame 
Buildings, is a special case of the last formula. 



A ' Fl^ 

10 E 



* See "Apparent Combined Stresses," Mer- 
riman's *' Mechanics of Materials." 



MECHANICS OF MATERIALS 



85 



TORSION. 

Circular Sections. . 
Twisting moment, M — Fa. 

Circular Secfiona 




dA. 



Fig. 60. 



Fig. 61. 



Resisting moment 



. M,= / 



-^SdA, where 



S is the shearing stress at the extreme fiber. 
M = M^, or 

Sh 



M = 



R 



where 7o is the polar moment of inertia. 

For a solid round shaft -5 = '^^^^ hence 






or aS = 



2_M 

7r223' 



Non-Circular Sections. (Taken from Mer- 
riman's "Mechanics of Materials.") For 
non-circular sections the above formulae are 
only approximate. 

For an elliptical section whose major axis is 
m and whose minor axis is n the maximum 
stress is 

16 Fa 



S = 



M = 



Trmn^ 

-n-mn^S 

16 



or 



yj rn 



86 MECHANICS OF MATERIALS 

For a rectangular section whose long side is 
m and whose short side is n, the maximum 
stress is 

^ 9 Fa 

2 

M= -mn^S. 

Transmission of Power. The horse-power 
which is transmitted by a shaft is 
2xa»F-a; 
• * 550X12 ' 
where a = moment arm in inches, 

a} = nimaber of revolutions per sec. 

OT 

But Fa = -^ , hence 

K 



ELLIPSOID OF STRESS. 

For any point within a stressed body, the 
resultant unit stress upon any plane is propor- 
tional to the radius vector of an ellipsoid. 
The principal axes of the ellipsoid coincide 
with the principal stresses, which stresses are 
normal to the planes upon which they act. 
For a plane not normal to a principal axis the 
resultant stress is not normal to the plane. 



REINFORCED CONCRETE. 
Notation. 

Let A = area, h = width of beam, h' = width 
of stem,.d = depth to center of steel, E = modu- 
lus of elasticity, / = unit stress, M = moment, 
n = Eg -i- Ec,P = total load, p = ratio of area of 
longitudinal steel to area of section of member, 



MECHANICS OF MATERIALS 87 

g = ratio of volume of circumferential steel to 
volume of column, s = spacing, subscript (c) 
refers to concrete, subscript (s) refers to steel, 
< = thickness of flange, u = umt bond stress, 
F= total shear, 2j = unit shearing stress, So = 
sum of perimeters of bars, and other values are 
as indicated in the figures or as specifically- 
stated. 

Columns. 

For columns with longitudinal steel only. 
P = f,A[l+{n-l)p]. 

fs = nU 

For columns with spiral and longitudinal steel, 
the proper form of equation is not well estab- 
lished. The ultimate unit load may be ex- 
pected to be 

in which fa is the yield point of longitudinal 
steel, fg' is the yield poiat of the circumferential 
steel, and X is a factor the value of which will 
usually be between 1.0 and 1.5 

Beams. 



For beams, in general, 

f, = M^(As3d). 

•'" n(l-/c) 
jd=d—z. 
v=V^(Jb'jd). 
u = V^{jdi:o). 

Per vertical stirrup, 

P = Vs/jd, 
Per stirrup at 45°, 

P=0,7Vs/jd. 







Fig. 62. 



88 MECHANICS OF MATERIALS 

For rectangular beams, 

k = ^2 pn-\-{pn)^—pn. 
z=kd-irS. 

f,=2M/jkbd^=2pfJk. 
(For /« = 16,000and/, = 650, p=0.0077.) 

(j is approximately ^-J 

For T-beams, 

2ndAg+bt'^ 
'^'^~2nA,+2bt' 
(3 kd-2 t) t 
^~ {2kd-t)^ 

(For thin flanges z is often assumed -•] 
For beams reinforced for compression. 



A.J^4'.L 




Fig. 63. 
k=^2n (p+p' f )+n2 (p+pO^J^-n (p+p'). 

^kH^-2p'nd' {k-^ 

k'^+2p'n{k-^ 

For p'=0.5 p, d' -J- d=0.10, and n = 15; 

p =0.0135 to make/s 16,000 when /„ is 747, 
for which M = 185 6d2; p=0.010 to make 
/« 16,000 when /^ is 650, for which M^ 
140 bd\ 



MECHANICS OF MATERIALS 89 

For p' =0.5 p, d' -T-cZ =0.15, and n = 15; 

p =0.013 to make /« 16,000 when /^ is 747, 
for which M=175 hd^\ p =0.0094 to make 
/« 16,000 when f^ is 650, for which M = 
130 ha'K 

For p'=p, d' -7-d=0.10, and n = 15; 

p =0.0195 to make /« 16,000 when /^ is 747, 
for which M=275 bd''-; p =0.014 to make 
/a 16,000 when /^ is 650, for which M = 
200 bdK 

For p' =p, d* -i-d =0.15, and n =15; 

p =0.018 to make /« 16,000 when f^ is 747, 
for which Af =250 6^2; p =0.0125 to 

make /, 16,000 when /^ is 650, for which 
■M =175 6^2. 

Flat Slab Floors. 

For flat slab floors extending over several 
panels in each direction, the following require- 
ments are in accordance with the recom- 
mendations * of the Joint Committee on Con- 
crete and Reinforced Concrete. 

Column Capitals. The minimum edge 
thickness should be 1| inches. The slope of 
the conical surface should not be more than 
45 degrees with the vertical. The minimum 
diameter (or dimension parallel to edge of 
panel) should be not less than one-fifth of the 
panel distance (measured center to center of 
adjacent columns), and it is desirable to use 
0.225 times the panel distance. 

Dropped Panels, The minimum width 
should be four-tenths of the panel distance, 
and the maximima ofifset should be five-tenths 

* For the Final Report of the Joint Com- 
mitttee on Concrete and Reinforced Concrete 
see Proceedings of the American Society for g g 
Testing Materials, vol. XVII, 1917, pp. 202- 
262. 



Zj z 



i 03 



90 MECHANICS OF MATERIALS 

of the thickness of the slab outside of the 
dropped panel. 

Slab Thickness. For, t = the total thick- 
ness of slab in inches, L = panel distance in 
feet, and w = the total dead and live load in 
pounds per square foot; minimum values of 
t should be 0.024 L v^ + 1| for slabs without 
dropped panels, 0.020 L V^j + 1 for slabs with 
dropped panels, and 0.03 L Vti; + 1| for the 
dropped panels themselves. Also, t should 
not be less than six inches, nor less than one- 
thirty-second of the panel distance for floors, 
nor less than one-fortieth of the panel distance 
for roofs. 

Bending Moments in Girdless Slabs. For 
c = diameter of capital in feet, and panel dis- 
tances in feet and in ac- ' 
/0» cordance with Fig. 64, 



I 

-! — -.^ 



.WL 



^<T^ and for other values as 
already given, interior 
panels may be designed 
upon the assumption 
that the sum of the posi- 
tive bending moments 
for one inner and two 
outer sections on one 
Fig. 64. ' ^i^® of length Li is ^ wLi 

(L2 — f c)2 foot-pounds, 
of which at least 25 per cent should be re- 
sisted by the inner section, while the two 
outer sections should resist at least 55 per 
cent of the positive moment in slabs with- 
out dropped panels, and at least 60 per cent 
in slabs with dropped panels. Also, for the 
slab thickness away from the dropped panels, 
at least 70 per cent of the positive moment 
should be resisted by the two outer sections. 
For interior panels, assume the sum of the 
negative moments to be resisted by one mid- 



MECHANICS OF MATERIALS 91 

section and two column-head sections along 
one line of length Li to be ^ wLi (Jj2, — | c)2 
foot-pounds, of which at least 20 per cent 
should be resisted by the mid-section; while the 

two column-head sections fof length -j, eachj 

should resist at least 65 per cent of the total 
moment for slabs without dropped panels, 
and at least 80 per cent for slabs with dropped 
panels. 

Wall Panels. At the first row of columns 
away from the wall and also at the sections 
halfway from this row of columns to the wall, 
increase the moments by 20 per cent of the 
values as determined for interior panels. If 
wall girders or cantilever restraint does not 
exist at the wall, increase the moments of the 
outer section and the column head section by 
20 per cent of the values as determined for 
interior panels, for designing reinforcement 
parallel to the wall. 

Shear and Diagonal Tension. As a measure 

of diagonal tension assume v = . , for slabs 

without dropped panels, and v = . for 

slabs with dropped panels. For punching 
shear at peripheries of capitals and dropped 
panels, assume a total shear 25 per cent greater 
than the actual punching shear, computed on 
the basis of a load which is uniformly dis- 
tributed. 

Bending Moments in Columns should be 
given special consideration. 



< g 



HYDRAULICS. 



NOTATION. 

A = area in sq. ft., a = area in sq. in., D = 
diameter in ft., ^ = energy, F = force, /= fric- 
tion factor, gf = 32.2, /i = head, ^y= friction head, 
7= moment of inertia, L = length in ft., M = 
statical moment, P = total pressure, p = unit 
pressure, g=quantity in cu. ft. per sec, r = 
hydraulic radius, s = slope, F = theoretical 
velocity, i? = actual velocity, ty = density of 
water. 

STATIC PRESSURE. 

p = wh, in which for water at ordinary tem- 
peratures w is 62.4 lb. per cu. ft. The density 
of water for particular temperatures is shown 
in Fig. 65. 











I3R2-":; I.-^S^- 




2 0-6 ^ ■s ^ 








Si -- 






















s 




^ 


5 fift 














j^ 50 100 150 200 

•^ Temperature deg. Fahr. 

Fig. 65. 

For h in feet, 

p = 62.4 h lb. per sq. ft. 
= 0.433 h lb. per sq. in. 
or /i = 2.306 p, for p in lb. per sq. in. 

P = pA 

= I whdA, for any surface, 

= whA, for a horizontal surface, 
92 



HYDRAULICS 



93 



or = ^ whA, for a rectangular surface with 

one edge at the surface of the water 
h being measured to the lower edge. 

If p is the average unit pressure, 

Py = Pcosd = pAf,. 

Ph = Psmd = pAy, 

CENTER OF PRESSURE. 



= ArQ^-^Ay, 

in which r is the radius of 
gyration. 

WEIRS. 




Fig. 67. 




Fig. 68. 



q = cu. ft. per sec. 

h = observed head in 
ft. 

V = velocity of ap- 
proach. 

Velocity head, 



K = 



2g 



Rectangular Weirs. 

Francis^ formula is 

Q = 3.33 [L - 0.1 nh] [(h+h^)^ - h,^, 

in which n is the number of end contractions. 

Fteley and Stearns' formula for suppressed 
weirs is 

q = 3.31 Lih + 1.5 h,)^ + 0.007 L. 

Bazin's formula for suppressed weirs is g = 



(0.405+5:5^^)[l+0.55( 



H+h 



)']LV2^.Ai. 



< a 

lu Z 
I "J 



94 



HYDRATJLICS^ 



A general equation for discharge is 

q = c%^2j'L(Ji+nhy)% 

for which Hamilton Smith 's values of n are 1.4 
for contracted weirs and Ig for suppressed weirs. 




0.5 1.0 
Head.Aaii feet 

Fig. 69. 

Contracted Weirs 



0.5 1.0 ( 
Head.Ti in feet 



Fig. 70. 
No End Contraction. 
Smith's values of c for contracted weirs are 
and for suppressed weirs in 

Trapezoidal Weirs. 

Cippoletti^s formula is 
q=-S.367Lh^, 
Triangular Weirs. 

Q 

q== c » ~- tan e ^2g'h^, 
lo 



plotted in Fig. 
Fig. 70. 



Fig. 71. 
Cippoletti Weir. 




For 6 = 45°, and for an 
■^ average value of c, 
Fig. 72. q = 2.6h^. 

Triangular Weir. 

Submerged Weirs. 

The formula for submerged weirs given in 
the American Civil Engineers* Pocket Book* is 

* American Civil Engineers' Pocket Book, 
page 854. 



HYDRAULICS 



95 



q = cLV2^Jh-^ih-h')y 
in which c is from 0.58 to 0.63 for a sharp crest. 




1 — - 




^^ 


g ^. 


\ 


"o5 V 


gO.5 r^ 


:! 4 


> ^ 







0.5 1.0 

Tlatio h-^h 

Fig: 73. Fig. 74.* 

Submerged Weir. 

HerscheVs formula* is 

Q = 3.33L(n/^)f, 
in which n has values indicated in Fig. 74. 

ORIFICES AND JETS. 
Discharge. 

V2 



V=V2gh or h = 



2g 




i-3D^ 



W^ 



Fig. 75. 
Standard Orifice. 



Fig. 76. 
Standard Tubt. 



For a standard orifice, 

v = iTom 0.97 V to 0.99 V. 
A' = from 057 A to 0.62 A. 

in which an average value of c is 0.61. 

* Trans. Am. Soc. C. E., 1885, vol. XIV, 
p. 194. 



" 6 
si 



96 HYDRAULICS 

For a standard tube, 

2j = 0.82 7 = 0.82>/2^, 
g = 0.82A>/2^. 

An inward projecting tube may reduce the 
discharge to 0.5 A '^2gh, and a diverging or 
compound tube will increase the discharge. 

Force and Energy. 
The energy of a jet discharging W lb. of water 

isTF— . 
2g 

The f.orce of a jet discharging W lb. of water 

per second, and impinging at right angles to a 

V 

fixed plate, is TF - • 
g 

The impulse exerted by a jet is equal to the 
reaction. For a jet 
R deflected by a fixed 



^3 



Fi 




plate, 

g 
fi = 2Fisin| 

= 2TF%in|. 

The total component of force parallel tc 
Fi (Fig. 77) is 

Fi-Fa cos 0= TF - (1 -cos Q). 
Q 

For moving plates the force upon the plate is 
that which would be exerted upon a fixed plate 
by a jet having a velocity equal to the relative 
velocity of the given jet to the plate, and the 
work done can be computed from the forces 
(impulse and reaction) and the velocity of the 
plate. The velocity of the plate determines 
the distance through which the force acts. 



HYDRAULICS 



97 



Also, the energy given up by the jet may be 

W 
computed by the formula ^r— {jo-^ — x><^), in which 

»i is the absolute velocity with which the jet 
strikes the plate and X}^ is the absolute velocity 
of the jet leaving the plate. 

FLOW IN PIPES. 
Long Pipes. 
Fanning' 8 formula is 



hf=4.f^ 



L 2J2 



D2g 



-r-^' 



o-oiirmr 














^'> 


A A-JO ^ 








\ 






^ 3j!rf , 




^^4?f 






U.UIU r - - 




r^ 






3;^7t 






®ooos i'dl 






f^cf^i 


B 41 


c3 ^ 1 ! 




> v.l'--^ 


.-Xejp^ 


ft Oftfi N^^ ' 






%^a!!. 


^^ 


to 


*^^ 


ftom. '^ "^ 


i i>-~ 


se^Tri 




bO*'> 


1 -1 1 1 


1 { 1 


0.00-;? 1 1 ii 


IN • 



5 "10 

Vel..in ft. per sec. 

Fig. 78. 




Vel. in ft. per .sec. 
Fig. 79. 



whence 






Values of / for cast iron pipe are indicated in 
Fig. 78.* (The formula for hf is frequently 

* Plotted for average values given in Ameri- 
can Civil Engineers' Pocket Book, pp. 845, 846. 



98 HYDRAULICS 

used in the form /y=/— jr— , in which case the 

value of / is four times the value here used.) 
Chezy's formula is 

in which r is the hydraulic radius, which is equal 
to Z)/4 for a pipe, s is the slope of the hydraulic 
grade line or the friction head divided by the 
length, and C is a coefficient, for which values 
for cast iron pipes are indicated in Fig. 79. 

The Chezy formula is used for flow in open 
channels as well as for flow in pipes. 

FlamanVs formulce are 

2J = 86.38 D7s7 for new cast iron pipes, 

V = 76.28 Z)7s7 for old cast iron pipes,] 

in which s is the same as for the Chezy formula. 

Various Losses of Head. 
The loss at entrance for a pipe is 

in which Cy is the coefficient of velocity. 

For a square edge at the entrance, the loss 

may be taken as 0.5 ^r— , or for an inward pro- 
^ g 

2j2 

jecting pipe it may be considered to be jr— • 

^ 

The loss due to expansion at a point of sudden 

enlargement is 

in which i?i is the velocity in the smaller section 
and xii is the velocity in the larger section. 

Other losses of head occur at elbows, valves, 
and sudden contractions. These are ordinarily 

stated in the form i? jr— , in which K isa, coeffi- 



HYDRAULICS 99 

cient for the particular case. For practical 
problems the equivalent length of pipe may 
often be used. 

Equivalent Pipe Length. 

A convenient method of taking account of 
losses of head at entrance, elbows, curves, and 
fittings, and the head remaining as velocity- 
head ( o— ) is to add to the actual length of 

pipe a length in which the friction loss would 
be equivalent to the particular loss, using the 
total equivalent length of pipe in computing 
size or flow. The equivalent length of pipe 

required to produce any loss, K — , is — times 

the diameter, in which / is the friction factor 
for Fanning's formula. For ordinary compu- 
tations for iron pipe the following equivalent 
lengths may be used: 

For loss at entrance, 25 diameters, 
For loss at an elbow, 10 diameters, 

For loss at end, ( tt- I . 50 diameters. 



Bemoulli*s Theorem. 

Neglecting friction, P/iy + i)2/2 g-\-z is a con- 
stant for all points along a given pipe, z 
being the elevation of the point above a given 
plane of reference. 

FLOW IN CHANNELS. 

Chezy's formula is 

v = C^rs, 

in which s is the slope, r is the hydraulic radius, 
which is equal to the area of the cross-section -^ 
of the water divided by the length of the 
wetted perimeter, C is a coefficient which de- 6 
pends upon the roughness of the channel, and Hi 
V is the mean velocity. 



o 



100 



HYDRAULICS 



Kutter's formula for the value of C for use in 
the Chezy formula is 



1.811 , ,, ^^ , 0.00281 
h41.65H 



C = 



1 + 



-f 
v9V 



4l.65 + 2:22?81) 



in which r and s are the same as given for the 
^^_^_^^ Chezy formula, and n 
^^^^^Pj is the coefficient of 
roughness for which 
some of the values are 
as follows: 

71 = 0.010 for neat 
cement, 

n = 0.013 for clean 
brick and sew- 
ers, 

n = 0.015 for unclean 
sewers, 

n = 0.020 for new 
canals, 

71 = 0.025 for ordi- 
nary canals, 

n = 0.035 for canals 
in bad condi- 
tion. 

Values of C for 7i = 0.015 and n = 0.025 are 
indicated in Fig. 80, for s = 0.01 and for s = 
0.0001. 

Bazin's formula for the value of C for use in 
the Chezy formula is 

157.6 




1 5 10 

Hydr. Bad. in ft. 
Fig. 80. 
Kutter's Coefficient. 



1 + 



1.811m 



v; 



or 



87 



0.552 + -^ 

Vr 



HYDRAULICS 101 

in which r is the hydraulic radius and m is a 
coefficient of roughness for which some of the 
values are: 

w = 0.16 for planks or bricks, 
w= 1.30 for ordinary canals, 
m=1.75 for canals in bad condition. 

The ratio of the mean velocity in a channel to 
the maximum surface velocity is subject to a 
considerable variation, its approximate value 
being 0.8. 

The ratio of the mean velocity for any vertical 
section to the velocity at the mid-depth is approxi- 
mately 0.98.' 

For any vertical section, the velocity at 0.6 
of the depth from the surface will be approxi- 
mately the mean velocity for the section. 

For any vertical section the mean velocity is 
. approximately 0.9 of the surface velocity. 

HYDRAULIC GRADE LINE 

The hydraulic grade line is the line connect- 
ing the points to which water would rise in a 
piezometer tube, if the tube were applied to 
consecutive points throughout the length of a 
pipe or conduit. The distance from the pipe 
to the hydraulic grade line at any point is 
the pressure head at the given point. The 
slope of the hydraulic grade line is the hydrau- 
lic gradient. The difference in elevation 
between any two points on the hydraulic grade 
line is the loss of head which exists between 
the two corresponding points of the conduit. 



m Z 
I "J 






H -J 



HEAT ENGINEERING. 

Compiled by G. A. Goodenough 

Professor of Thermodynamics, University 
of Illinois 



ELEMENTS OF THERMODYNAMICS. 

Notation. 

M = weight of substance, in lb. 

p = absolute pressure, in lb. per sq. ft. 

t = temperature, deg. F. 

T = t + 459.6 = absolute temperature. 
V, V = volume, in cu. ft. 
TJ , u = internal energy, in B.t.u. 
7, i = thermal head, in B.t.u. 
S, s = entropy. 
Q, q = heat absorbed in B.t.u. 

J = 777.6 = mechanical equivalent of heat; 
i.e., 777.6 ft. lb. = 1 B.t.u. 

A = — = reciprocal of mechanical equiva- 
lent. 

W = external work done during a change of 
state. 

Cj, = specific heat at constant volume. 

Cp = specific heat at constant pressure. 

The small letters, v, u, i, s refer to 1 lb. of the 
substance, the capital letters V, U, I, S refer to 
M lb. Thus V = Mv, S = Ms, etc. 

Fundamental Equations: Definitions. 

The state of a substance initially given by 
Ph ^1. Ti changes to a second state given by 
P2> ^2, T2. The work done by the substance in 

pdV; and if Q12 denotes 

the heat absorbed during the process, the first 
law is expressed by the energy equation 

V2 
Qj2 = U2-Ui+A I pdV, 



102 



f 



HEAT ENGINEERING 103 

Or in differential form, 

dQ =du +ApdV. 
The thermal head I is defined by the equation 
I = U + ApV, 
whence dQ = dl — AV dp. 

For a change of state at constant pressure, 
dQ = dl, or 

Ql2 =I2- h- 

Similarly, for a. change of state at constant 
volume 

Qi2 = U,- U,. 

The entropy S may be defined by the relation 
• „ dQ , dH 

where H denotes, not the heat absorbed by the 
substance from the surroundings, but the heat 
generated within the substance due to friction, 
wire drawing, etc. If the change of state is 
adiabatic (no heat absorbed or rejected), then 

dQ = and ds = -^ • If the change is also 

frictionless, dH = 0, and dS = 0, or S is con- 
stant. In many changes H is negligible, whence 

ds^^, or dQ =TdS, 
Qu = I 'TdS. 



s 



It follows that if the change of state is repre- 
sented graphically on a plane with T and S as 
the axes, the area between the curve and the *S- 
axis represents the heat absorbed. 

PERFECT GASES. 

The characteristic equation of a perfect gas ia < 5 
pv = BT, or pV = MBT, 

in which B is the so-called gas constant. The 
equation may be given the homogeneous form 

P\V\ ^ V2V2 _P^ _ jy 
Tx Tz T 



z 

I "J 



— ' l.l 



104 



HEAT ENGINEERING 



For a perfect gas 



c^-c, =A5=^^^ 



B 



and Cp/Cy = k. 

VALUES pF B, Cp, c^, AND A: FOR GASES. 



Gas. 


B 


^P 


c. 


k 


Air 


53.34 
765.86 
54.99 
48.25 
55.14 


0.240 

3.42 

0.247 

0.217 

0.243 


0.171 

2.44 

0.176 

0.155 

0.172 


1.40 
1.40 
1.40 
1.40 
1.41 


Hyd.roo"en 


Nitrogen 


Oxygen 


Carbon monoxide. . 



For a change of a gas from an initial state 
Vii ^1. Ti, to a final state P2. ^2. 2^2. 



U2-Ui=Mcy{T2-Ti) = 
h-h^ Mcp (^2 - Ti) = 



k -1 

Ak 
k -1 



(P2V2-PlVi), 



S,~Sl^ M ^cploge —^ + c,log, ^^]. 



special Changes of State. 

1. Constant Volume. ^ = ^. Wu = 0, 

Qi2 = U2-Ui= Mcy {h - ^1). 

rp 

/S2 - 5l = MCy lOge ^T • 

2. Constant Pressure. -^ = -p^ . 



Q] 



Afc„ («2 - ^1) = 



Ak 



k -1 



Wx., 



S2 - Si = Mcp logg ~ • 



HEAT ENGINEERING 105 



3. Constant Temperature (Isothermal). 

PiT^i = P2V2. W12 = PiVi log, Y ' 

U2 - C/i =0. 
Q12 = AWv, = AMBT loge ^• 

S2-S1 =^=AMBloge ^- 

TF12 = J (L^-C72)= ^^ (PiFi - P2V2). 
Q12 =0. ,S2 - Si = 9. 



5. pv^ = const. The expansion and com- 
pression of gases in motors, compressors, etc., 
may be represented by curves having the equa- 
tion pv^ = C, where n is a constant. The 
specific heat associated with such a process is 

Cn = ^v T . whence for 1 < n < k, c^ is 

71 JL 

negative. 

n— 1 n—\ 

n \vj \vi) 



=f^l[^-(r] 



Q12 = Mc^ {h - h) S2-Si = Mc„ loge ^1 • 
AWi2 :U2-Ui:Qi2=k -I :l -n:k -n. 



106 HEAT ENGINEERING 

SATURATED AND SUPERHEATED 
STEAM. 

notation. 

The symbols v, u, i, and s have the same sig- 
nificance as in the general notation; however, 
these symbols with a prime iv', s\ etc.) refer to 
1 lb. of water at the boiling temperature, and 
with a double prime {v'\ u", etc.) they refer to 
saturated steam. In addition, let 

r = latent heat, i.e., heat required to vaporize 
1 lb, of liquid at given constant pressure 
and temperature. 

tp = Ap {v" — v') = heat equivalent of ex- 
ternal work required in vaporization. 

p = increase of energy during vaporization. 

X = quality of mixture, i.e., ratio of weight of 
steam present to total weight of mix- 
ture. 

c* = specific heat of water. 

Fundamental Relations. 
i" = i' + r. u" =u' +p. r ^ p +yp. 

T 



, f ^ , dT 

•^ 491.6 







^ = A {v" — ■»') "jT * (Clapeyron*s relation.) 

For a mixture of steam and water having a 
quality x, 

\ = i' + xr = i" — (1 — x) r. 

u = u" — {I — x) p. 

s =s'+a:^ =s"-(l -x)^' 
» =s c' + a; (v" — v') = xv" approx. 



HEAT ENGINEERING 107 

Equations for Superheated Steam. 

[In the following equations, take p in lb. per 
sq. inch.] 

„=:^-(l+3api)^+0.018. 

Cp = 0.32 + 0.000126 T + ^^ 
+ p(l+2ap^) ^. 

i = 0.32 r + 0.000063 T^ - =^^ 
-p(l -{-2ap^)~ +948.54. 

1 1700 
s = 0.73683 log T + 0.000126 T - ^^^ 

- 0.254 log p - (1 + 2 ap^) ^ 

- 0.0807. 

u = i — 0.1852 pv. 
Constants in the preceding formulas: 
log B = 1.77448. log C = 11.39361. 
log 3 a = 2.71000. log C" = 10.79155. 
log 2 a = 2.53391. log C" = 10.69464. 
log m = 10.82500. 

Tables of the Properties of Steam. 

Two tables of the properties of steam are in- 
cluded among the tables of this book. The 
first gives the important properties of saturated 
steam, while the second gives properties of 
superheated steam and also of mixtures of steam 
and water within certain limits. In this second 
table values of the entropy from 1.50 to 1.85 
inclusive appear at the top of the page, and 
values of the pressures are given in the first 
column. By following a column the variation 



< ^ 



108 HEAT ENGINEERING 

of the volume v and the thermal head i during 
an adiabatic change of state is observed. The 
column designated by x gives the temperature 
of the superheated, steam above the heavy 
dividing line and the quality of the mixture be- 
low this line. 

Changes of State in Steam and Water 
Mixtures. 

1. Isothermal or Constant Pressure, t — 

const, p = const. 
Wi2=P (V2 - Vi) = Mp (v" - v') (X2 - Xi). 
U2 - Ui = Mp (X2 - xi), Q12 = Mr (X2 - ^1)' 

2. Adiabatic. s = const. 

Wn = ( Ui- U2) J = JM [(^V + x,pO - (iV+ X2P2)] . 

3. Constant Volume, v = const. 

ft 

XiVi" = X2V2". or X2 = xi -^,' 

Wn = 0. Q12 = C/2 - C/i 

= M [{i2+X2P2) - (iV + XiPi)J. 



FLOW OF COMPRESSIBLE FLUIDS. 

Fundamental Equations. 

Let A denote the cross-section of the pipe or 
tube through which the fluid is flowing, lo the 
mean velocity of the fluid across the section, 
and M the weight in lb. flowing through the 
section per second. If the flow is adiabatic, as 
may usually be assumed, the following equa- 
tions apply. 

1. Equation of continuity. 

. _^ AiWx A<iW^ 

Aw = Mv, or — - — = — =— = • 



HEAT ENGINEERING 109 



2. Equation of energy. 

J* + ^T- = const., or Ji + ir- = Ji2-r ir" 
Z g Zg Zg 

The second equation may be expressed by the 
statement: the sum of the thermal head and 
the velocity head is a constant. 




The two equations hold good for flow with 
friction. The effect of frictional resistances is 
to increase the thermal head Ji and decrease the 

velocity head ;r— by an equal amount. 
■^ Q 

In the case of flow from a reservoir, as a steam 

boiler, the initial section A\ may be considered 

inside the reservoir and the velocity w\ may be 

neglected in comparison with the exit velocity 

\ W2, In this case the second equation becomes 



= J (ii — ii) 




or w = ^2gJ ^ {ii - i^) = 223.7 ^{ii - I'a). 
For air, or other gases of similar nature. 

For steam, values of i are given in the steam 
tables. 

Discharge Through Orifices. 

Let Pi denote the pressure in the reservoir, po 
the pressure in the region into which the fluid 
discharges, and P2 the pressure in the plane of 
the orifice, that is, at section Ao. If Po is less 
than a certain critical value mpi, then P2 takes 



110 HEAT ENGINEERING 

the value mpi, and the discharge is constant for 
ail values of po^ If. however, po is greater than 
^Pu P2 = Po. and the dis- 
charge decreases as po ap- 
proaches pi. The value of 
m depends upon the prop- 
erties of the fluid. For sat- 
urated or slightly wet steam 
m = 0.58; for superheated 
steam m =0.55; and for air 
and similar gases m ~ 0.53. 
Case 1. Po ^mpi. Dis- 
charge is independent of po. 
P2 = mpi. Take the flow as frictionless and 
find 11 and 2*2 corresponding to pi and P2 from 
the second of the steam tables. Then 




W2 = 223.7 VHi - iz) and M = ^ • 

For example, steam at 190 lb. pressure super- 
heated to 450° flows through an orifice | inch in 
diameter into a region in which the pressure is 
60 lb. P2 = 0.55 pi = 104.5 lb., and from the 
steam table ii = 1241, 1*2 = 1189 B.t.u. and 
V2 = 4.26 cu. ft. Also A = 0.1104 sq. in. - 
0.000767 sq. ft. 



W2 = 223.7 V 1241 - 1189 == 1613 ft. per sec. 

^^a0007|7pi3^„_29 1b.per.sec.. 
or 17.4 lb. per min. 

For saturated steam, the discharge may bo 
calculated approximately by one of the follow- 
ing empirical formulas: 

1. Napier's rule, M = -=-r- • 

2. Grashof's formula, M = 0.0165 Ajfi-^. 

3. Rateau's formula, 

M = ^ (16.367 - a96 log p). 



HEAT ENGINEERING HI 

In these formulas p should be taken in lb. per 
sq. inch and A in square inches. Then M will 
give weight discharged per second. 

For the discharge of air with po < 0.53 pi, 
Fliegner's formula, 

ilf«0.53 ^. 

y/rp 

may be used. 

Case 2. po > ^Pi» P2 = 2'0' The discharge 
depends upon po and Pi. 

For steam, determine 2*1 and I'o, also 2Jo» then 

W2 = 223.7 V(ii _ io), M =^. 

For air the discharge in this case is given by 
the formula 



^=2.o5..\/A©»-V(r 



1. 



For small differences of pressure the discharge 
of air is given approximately by the formula 



M = 1.1 ^ 



V|(Pi-P*>- 



Vl^ 



Diverging Nozzles. 

Diverging nozzles are used when the back 
pressure po is less than the critical pressure 
Vm — ^Pi' The pressure at the smallest sec- 
tion, or throatf 
takes the value 
p^, and if the 
nozzle is prop- 
erly propor- 
tioned, the pres- 
sure at the end 
section A2 is po, 
the back pres- 
sure; i.e., P2 = Po* Taking Wj = in the reser- 
voir, the second general equation gives 

4g ig 



n 



112 



HEAT ENGINEERING 



If the flow is adiabatic and frictionless, the 
entropy remains constant and the three thermal 
heads i] , i^, and io are found in the second steam 
table. The effective drop of head through the 
nozzle is ii — iq. The effect of frictional re- 
sistances is to decrease this drop by y (ii — iq), 
where y is a coefficient that may vary from 0.08 
to 0.20 depending upon the size and smooth- 
ness of the nozzle. 





Thermal 
Head at 

End 
Section. 


Quality at 

End 

Section. 


Volume 
at End 
Section. 


Without 
friction. 

With_ 
friction . 




^0 


^0 

Vo'^XoW 



As an example, consider the design of a nozzle 
to discharge 0.7 lb. per second. The steam is 
initially at a pressure of 200 lb. per sq. in. and is 
superheated to 548° F. ; and the back pressure 
is 40 lb. per sq. in. The coefficient y is taken 
as 0.14. From the second steam table, in the 
column s = 1.65, the following values are found: 
ii = 1295, i,^ (for p^ = 110 lb.) = 1235, io = 
1150, v^ = 4.58, vo ^ 10.28. The loss of jet 
energy due to friction is 0.14 (1295 — 1150) = 
20.3 B.t.u. Without friction % at section Ao is 
from the table 0.978, with friction it is 0.978 + 
20.3 
935.5 

tion A2 is 10.51. Using the fundamental 
equations, the following results are obtained: 



1.00; hence the specific volume at sec- 



J 



HEAT ENGINEERING 



113 





1295 


1235 


'■'Fo'- 


1733 


Without friction.. . 
With friction 


1295 


1150 
1170.3 


ii - io 
145 

124.7 


Wq 

2694 
2498 




4.58 


A^ (sq. ft.) 
0.00185 


(inch) 
0.582 


Without friction... 
With friction...'... 


0.978 
1.00 


^0 

10.28 
10.51 


(sq.ft) 

3.00267 

3.00295 


^0 

(inch) 
0.700 
0.735 



Flow of Gases and Vapors in Mains. 

; The general equation of flow in pipes of circu- 
I lar cross-section, assuming that there is no 
I transmission of heat is 



vdp -{- -3- dL 

a 



0. 



in which d denotes the diameter and L the 
length of the pipe, and c is the coefficient of 
resistance. 

If the drop of pressure is small, as is the case 
in short mains, this equation gives the approxi- 
mate relation 



P =Pi -P2=c 



vd 



(a) 



When, on the other hand, the drop of pressure 
is considerable, integration of the general equa- 
tion gives the relation 



p{' 



Pi' = 



32 M^BTL 

Ti^ ^ d^ 



(&) 



in which M denotes the weight of air flowing 
per second. 



00 ^ 
f3 



114 HEAT ENGINEERING 

1. Flow of Steam. Since the drop of pressure 
is small formula (a) is used. The coefficient c 
is not constant but varies with the diameter of 
the pipe. Taking the diameter in inches, and 
the length L in feet formula (a) reduces finally 

.'=0.00013l(l+M)^. 



or Af =87 



- r v'^ - \i 



In these formulas M denotes the weight of 
steam flowing in pounds per minute, and v the 
volume of a pound of steam at the mean pres- 
sure p' in lb. per sq. inch. 

2. Flow of Compressed Air. Let V denote 
the volume in cubic feet of free air at 70° F. and 
a pressure of 14.7 lb. per sq. in. flowing per 
minute. Since in the flow of compressed air, 
the drop of pressure is relatively large, formula 
(6) is used. By proper transformations it may 
be given the form 

with c « 0.003 (i -I- Mj . 

Here again d is to be taken in inches, L in 
feet, and pi, P2 in pounds per square inch. 

THE STEAM ENGINE. 

Ideal RanMne Cycle. 

Representing the changes of state on the T/S- 
plane (see Fig.), the medium receives. heat in the 
boiler and superheater during the processes 
1-2-3; the line 3-4 represents adiabatic ex- 
pansion in the cylinder; and the line 4-1 repre- 
sents rejection of heat to the condenser. 



HEAT ENGINEERING 



115 



Heat absorbed = 3i = I'a — ii. 

Heat rejected = qo = U — H- 

Heat available for work = Qi — Q2 — *3 ~ ^i* 



T 


2L^ 




i 


\ 




k 




4 


\ 


/ 


1 

i 






> 




1 

! 









Thermodynamic efficiency of cycle 



^3 -^4 



AT- 2546 

H - H 



- iVp (4 - ii) 



Steam required per H.P. hr. — ly j^ 

B.t.u. required to give 1 H.P. hr. 
2546 

Values of ii, iz, and i^ are obtained directly 
from the steam tables. For example, steam is 
furnished at a pressure of 190 lb. per sq. in. 
superheated to 450** F., and the condenser 
pressure is 3 in. of mercury. Then 

ii = 83 B.t.u. ^3 = 1241 B.t.u. h = 913 B.t.u. 

Heat available for work = 1241 - 913 = 328 

2546 
B.t.u.; steam consumption per H.P. hr. = -__, 

= 7.76 lb.; thermodynamic efficiency of cycle 
~ 1241 — 8S ~ 0-283; B.t.u. required to pro- 
duce 1 H.P. hr. = 7.76 (1241 - 83) = 8986 
B.t.u. 



h-i 



116 HEAT ENGINEERING 

EflBciency of the Actual Engine. 

Under the same conditions of operation, the 
actual engine transforms a smaller amount of 
heat into work per pound of steam supplied 
than the ideal Rankine engine. Let q-^ and Qa 
denote the heat transformed by the Rankine 
engine and the actual engine, respectively. 
The efficiency of the actual engine is defined by 
the relation 

This efficiency ranges from 0.50 to 0.80 in 
steam engines and steam turbines. 

Let Na denote the actual steam consumption 
per H.P. hr. Then 

.- 2546 ^R ^R. 

Na = = — ; or 77 = — - , 

and the heat required to give 1 H.P. hr. is 

.... .. 2546 

Na(is-ii) =——' 
'i'jR 

In the example preceding let the efficiency of 
the actual engine based on the ideal Rankine 
engine be 0.70; then the steam consumption is 
7.76 -7- 0.70 = 11.1 lb. per H.P. hr., and the 
heat required per H.P. hr. is 8986 ^ 0.70 = 
12,837 B.t.u. 

STEAM BOILERS. 

Let »i = thermal heat (heat of liquid) of 
water fed to boiler. 
^2 = thermal heat (total heat) of steam 

formed. 
M = weight of water evaporated per hour. 
Mf = equivalent weight of water evapo- 
rated per hour from and at 212° F. 
/ = factor of evaporation. 
H = rated horsepower of boiler. 



HEAT ENGINEERING 



117 



By definition a boiler horsepower is equiva- 
lent to the evaporation of 34.5 lb. of water per 
hour from and at 212° F. 



/ = 



H = 



Me 



M_e 

M ~ 

Me 

34.5 
fM. 



971.7 



M (h - 12) 
33,520 



CONDENSERS. 

Steam enters the condenser at a known pres- 
sure pi and a quality Xi, which is frequently 
assumed as 1. The thermal head of the enter- 
ing steam is ii = ix + xiri\ that of the con- 
densed steam leaving the condenser at the tem- 
perature h is i-/. If M lb. of condensing water 
is required and the temperature at entering and 
leaving are ^3 and U, respectively, then 



M = 



k -h 



INTERNAL COMBUSTION ENGINES. 

The ideal cycles employed for internal com- 
bustion motors are the following: 

1. Explosive, Otto. 

2. Slow burning, non-explosive. Joule or 
Bray ton, Diesel. 



Otto Cycle. 




Compression 
1-2 and expansion 
3-4 are assumed to 
be adiabatic. The 
line 2-3 represents 
the rapid heating 
at constant volume. 



o a 



H-J 



118 



HEAT ENGINEERING 



—- /,,vAr-l 



Ti T4, \vJ \vJ \vj 

Heat absorbed = Qi = Mcy (T3 - T^). 

if Qi is the heating value of the fuel and M 
the weight of the charge of fuel and air, the final 
temperature T^ is 

^' = ^» + ]fe- 

The eflficiency of the cycle is 



Work of cycle = TT = vQi 
= JMc, (T3 



T,-T2 + T,). 



Joule or Brayton Cycle. 
The absorption of heat in the process 2-3 is 
at constant pressure. 




f=t=(^-fe^-eF 



McATz-TO; 1\= T,+ 



Mc^ 



Efficiency =17=1— 7=-=! 
^2 



m 



k 



Work of cycle = 17Q1 = JMcj, iT^-Ti-T^-h Ti\ 



HEAT ENGINEERING 



119 



Diesel Cycle. 

Air is compressed to a pressure of 500 lb. per 
sq. in. or more, and the fuel injected into this 
air burns at nearly constant pressure. 




£2 ^ (P2\ 

Ti \pj 



T2-^ 



Qi 

MCr. 



Tz \vz) 

7^4 -Ti 



Efficiency = 17=1 -^^y^_^^^ 

Work of cycle = 
-nQ, = JM [Cp (Ta - T2) - c^ (^4 - Ti)]. 



AIR COMPRESSION. 

Let Vi = volume of free air entering compres- 
sor cylinder per stroke at pressure 
pi (atmospheric, or sUghtly lower). 

V2 = volume of the same air when com- 
pressed to the higher pressure P2' 

W = work required per strike. 

H = net horsepower required to drive the 
compressor. 

N = r.p.m. of double acting compressor. 

The compression is assimied to follow the 
law pv^ = const. The value of n lies between 



aj w 



J 



120 HEAT ENGINEERING 



1.2 and 14 depending upon the effectiveness of 
the water jacket. An average value is 1.3. 



H = 



(P2 and Pi in lb. per sq. foot.) 
2NW 
33,000* 



If the compressor has no clearance the volume 
Vq swept through by the piston is equal to V\. 
If there is clearance, the air caught in the clear- 
ance space expands from pi to pi and as a result 
V\ < Vq, Let m = ratio of clearance volume. 
to Vq\ then 

y„ li 



l+»-»(D" 



Compound Compression. 

If the air is compressed in two stages (1) from 
Pi to an intermediate pressure p' (2) from p' tc 
P2» then for minimum work of compression 

p' = ^pipi 

and ,r = ;^...[gf -1} 

For compression in three stages with cooling 
between the stages, the proper intermediate 
pressures are 

p' = ^Pi2p2, V" = ^M?, 
and the work of compression per stroke is 



HEAT ENGINEERING 



121 



REFRIGERATION. 

Air as the Medium. 

Air is compressed adiabatically, as shown by 
1-2, cooled at constant pressure (2-3), expanded 
adiabatically (3-4) in a separate expansion 
cylinder, and then passing through the brine 
absorbs heat from it, as represented by 4-1. 



M 
N 


i^i 


J 3 


2 




V 


1 


^_ 






4: 


1 


T 


) 


2 

Jemp. of Water 








'jTemp. of Brine 




1 




1 



Let Q = heat absorbed from brine or cold 
room per minute. 

Q' = heat rejected to cooling water per 
minute. 

M = weight of air circulated per minute. 

H = horsepower (net) required. 
Pi. P2 = lower and higher pressures, re- 
spectively. 

Cp = specific heat of air at constant 
pressm-e. 

The temperatures Ti and Tz are fixed by the 
brine and cooling water; the temperatures T4 
anci T2 are obtained from the relation 



122 



HEAT ENGINEERING 






Q = Mcp CTi - T,). 
Q' = Mcp (Ti ~ Ti). 

Work per minute = J (Q' — Q) 

= JMcp [{T2 - Tz) - (Ti - TCi\ 

J., _ Q T,-T, 
^ 42.43 Ti 
If N is the number of working strokes per 
minute, the required volume of the compressor 
cylinder (neglecting clearance) is 



V,= 



MBTi , 

Npi 



that of the expansion cylinder is 
MBTj 

Npi 



Ve = 



Vapor as the Medium- 
Adiabatic compression (1-2) is followed by 
rejection of heat (2-3) to the cooUng water 

i 




until the medium is a liquid (at point 3). 
Liquid passes through expansion valve dropping 



HEAT ENGINEERING 123 

in pressure from P2 to pi and attains state repre- 
sented by point 4, with 13 — 14. Line 4-1 
represents absorption of heat from brine at 
constant pressure pi. Using same notation as 
in preceding section, 

Q = M (ii - 14) = M {ii - $3), 
Q' = M (i2 - 23). 

Work required per minute = JM {i^ — ii). 

42.43 

Let v" denote the volume of 1 lb. of the 
saturated vapor at the lower pressure pi; and 
N the number of working strokes per minute; 
then the volume of the compressor cylinder is 
(neglecting clearance) 

^^ ~ AT * 

If the cooling water enters at the tempera- 
ture ti and leaves at temperature ^2» the weight 
G required per minute is 

^ ^ M (i2 - iz) 
h — h 

Values of I'l, 12, and 13 are obtained from the 
tables of saturated and superheated ammonia 
or carbon dioxide. 



3i 



ELECTRICAL ENGINEER- 
ING FORMULiE. 

Compiled by H. H. Higbib. 

Professor of Electrical Engineering^ University 
of Michigan, 



NOTATION. 

A = area, square centimeters. 
A^ = cross-section area magnetic circuit, sq. 
cm. 
B = flux density, maxwells per sq. cm., 
gausses. 
Bm, = cyclic maximum flux density, gausses. 
b = susceptance, mhos. 
C = capacitance, farads. 
Co = capacitance to neutral, per mile ot 

transmission line, farads. 
d = distance, centimeters. 

= diameter, mils. 
E = e.m.f., volts, effective or square-root- 
mean-square value. 
= unvarying voltage in d-c. circuit. 
^av = average value of varying e.m.f., volts. 
E^ = maximum instantaneous value of 

varying e.m.f. 
^0 = volts to neutral, r.m.s. value. 
Er = volts consumed in overcoming resist- 
ance. 
Eg = e.m.f. generated, volts. 
Et = e.m.f. between terminals, volts, 
c = e.m.f. at any instant, volts. 
F = force, dynes. 
/ = frequency, cycles per second. 
ff = magnetomotive force (m.m.f.), gilberts. 
124 



ELECTRICAL ENGINEERING 125 

g = conductance, mhos. 
H = magnetizing force, field intensity, 
gausses in air, dynes force on unit 
pole. 
I = current, amperes, effective or r.m.s. 
value. 
= unvarying current in d-c. circuit, 
1^ = current from terminals, amperes. 
If = current in (shunt) field, amperes. 
la = total current through armature, am- 
peres. 
i = current, amperes, at any instant. 
»c = charging current at any instant, am- 
peres. 
k = specific inductive capacity or dielec- 
tric constant. 
K = constant. 
L = self-inductance of electric circuit, 

henrys. 
I = length. 
Ijn = length magnetic circuit, centimeters. 
l^t, = length of wire, centimeters. 
M = mutual inductance, henrys, of two 
electrical circuits magnetically in- 
terlinked. 
m = power factor. 

= strength of magnet, in unit poles. 
n = reactive factor. 

= angular velocity, revolutions per sec- 
ond. 
N = turns in coil or electrical circuit. 
p = instantaneous power, watts. 

= number of field poles. 
P = average power, watts. 
Ffl- = power lost due to hysteresis, watts. 
Pe = power lost due to eddy-currents, 

watts. 
Py = power, watts, transformed into heat 
in overcoming resistance. 



"n z 



m TO 



126 ELECTRICAL ENGINEERING 

Q, Q= quantity of electricity, coulombs, am- 
peres X seconds. 
R = resistance, ohms. 
(5{, = reluctance, oersteds. 
r = radius, 
s = number of parallel paths between 

armature terminals. 
T = torque, pound-feet. 
i = thickness, thousandths of inch, mils. 
= temperature, degrees Centigrade. 
= time elapsed, seconds. 
V = velocity, centimeters per second. 
V = voliune, cubic inches. 
w = weight, pounds. 
Wm, = energy of magnetic field, watt-sec. or 

joules. 
Wc = energy stored in condenser, watt- 
seconds. 
X = reactance, ohms. 
y = admittance, mhos. 
Z = impedance, ohms. 

= number useful conductors on arma- 
ture. 
Ofl = temp, coeff. of resistance, based on 
0°C. 
6 = base of Naperian logarithms =2.7183. 
7] = efficiency, ratio. 
6 = angle. 
= time-phase difference expressed in 
electrical degrees. 
/i = permeability, ratio. 
Po = resistivity at 0° C, ohms per centi- 
meter cube. 
$ = magnetic flux, maxwells. 
CO = angular velocity, radians per second. 



ELECTEiCAL ENGINEERING 12i 



I 



MAGNETIC FORCES AND FIELDS. 

(a) Field due to a vole at a point. 



F<-^ 






Fig. I 

rrh) = 
exerted on unit north pole, 

(6) Field due to current in straight conductor 



H2 (at mi due to Ws) = — = ^orce (dynes), 



F = "^ = m,H, = m,H,. 

d^ ■ . 




H {at P due to i) = ^qT^ (sin^i + sin^a). 

Field (dynes force on unit north pole) at P 
is downward into paper if current flows toward 
right, and upward if current flows toward left. 
Field is circular and concentric with axis of 
conductor. 

(c) Force on conductor due to current and field, 

F (dynes) 

^ 

,, - p/Maxwells \ s^ 

Qpi diaperes 



p /Maxwells \ 




3. Conductor Fig. 4. Uniform Fig. 5. Conductor 
alone, field alone. in field. 



128 ELECTRICAL ENGINEERING 



F (dynes on each centimeter length of wire) 
= B — , whence 



Pounds force on wire 

= 22.5 X 10-8 

^5M 

108 



Bl^i 
Bi X (length of wire, inches). 



This formula presumes that i is in direction 
at right angles to B. If the directions of i and 
B form an angle d, the preceding expression for 
force must be multiplied by sin 6. This force 
is perpendicular to both i and B; it is in 
direction away from the side of the conductor 
where the field has been made more dense, and 
toward the side where the field has been made 
less dense (Fig. 5; . 

id) Law of the magnetic circuit. 
f7 _ 0.4 irNi 

(R" 



$ = 



C/V^> 






whence 



H=QA'ir 






and 

Amp.-turns per inch length of magnetic circuit 

_ / maxwells per square inch \ 

— \j.o\o2i X I I* 

\ M / 

6'ee page 150 for magnetization curves. 



MAGNETICALLY INDUCED ELECTRO- 
MOTIVE FORCE. 







J 




/ 


\ ^ 








; jF^i-^-^ 




i. : : : b^mj^wu) 








: : : : : ^ 




upward .thru 

paper. . 










' 


^ ^ 






- i 



Fig. 6. 



ELECTRICAL ENGINEERING 129 

I e (induced) =^^=Jq5-^, 

where $ is the total maxwells linking the single 
turn of circuit shown. Direction of e is always 
such that force produced on current in same 

j direction as e, by the field, would be in direc- 
tion opposite to the velocity which produces e. 

I In general, if N turns are linked by a varying 

I fltix * maxwells, then 

If a current i amperes flows, the conductor 
must move against a force ( -ttt ) dynes, whence 
Bxl^ ^, lO'F Fv dyne-cm. per sec 

^'= To^ ^ ^17 °^ ioi~ • °' 

volts X amperes = watts 

ergs per sec ^ . ^ ^ , , 
= ^ = 746 X horse power. 



INDUCTANCE OF AN ELECTRIC 
CIRCUIT. 

(a) General. An electric circuit has 1 henry 
inductance if 1 volt is induced in it when the 
current changes at rate of 1 ampere per second. 
j A non-inductive circuit is one which builds no 
magnetic field when current flows. The in- 
duced e.m.f. must always op-pose the change of 
current. 

. (induced) = -^ I = - 1^ I (*^)' 

e (average) = " ^qs >< ~' 

1 ^N 
L (average) = 3^ X — » 



aj cr5 



130 ELECTRICAL ENGINEERING 

where $ is the flux produced by i, which links 
all of the turns N, and L is the average in- 
ductance within the current limits to i, or 
€ux limits to $. If not all the flux links all 
the turns, but *i maxwells link Ni turns, #2 
maxwells link N2 turns, etc., we have 

L (average) = jQ-g X -. 

(6) Self-inductance of a transmission line in 
air, henrys per mile length of each single wire, 
is given by the equation 

L^ = (0.O8O47 + 0.74113 logio-) X 10-3, 

where d is distance between centers of out- 
going and return wires, and r is radius of wire, 
both in terms of same unit of length. Tables 
of inductance and reactance for transmission 
lines, found in Handbooks, are calculated from 
this formula; it applies also to each mile of 
each wire of a three-wire line if wires are all 
equidistant. 

(c) Mutual inductance of two electric circuits. 

nr dio , ,, dii 

€1 = — M --, and e^ = — ivf -r- . 

at " at 

^ J_ *2iVi ^ _1_ f^ 

^•^W 108 ' i^ 108 * i ' 

where ii amperes in one circuit cause *i max- 
wells to link with the iV2 turns of the other 
circuit, or io amperes in the second circuit cause 
<y>2 maxwells to link with the iV"i turns of the 
first circuit. If all of the flux produced by 
either of the two circuits links with all the 
turns of both circuits, we have: 
L1L2 = M2. 

ENERGY STORED IN MAGNETIC FIELD. 

Wm =hLi'' = f Lidi 
•^0 



ELECTRICAL ENGINEERING 131 

gives the watt-seconds or joules of energy 
stored in magnetic field due to current i am- 
peres in circuit having constant inductance L 
henrys. 

Sir fji Sir 

gives the ergs per cubic centimeter in a mag- 
netic field of density B maxwells per square 
centimeter in a medium having constant per- 
meability /i. 

POWER DISSIPATED IN MAGNETIC 
CIRCUIT. 

Ph = KJB.J-^ V = KJB^^'^ w. 
Pe = K,PB^HW = K^PBrr^H^w. 

Values of K, and of P^ or Pe for any assigned 
values of /, Bj^, V, w and t may be calculated 
from data given on pages 150, 151. It is 
assumed that the flux varies harmonically with 
respect to time, and that it is uniformly dis- 
tributed throughout the iron. 

CONDENSERS AND ELECTROSTATICS. 

(a) General. 

C = ^, OT Q =Ce. 
e 

C is in farads when q is in coulombs or am- 
pere-seconds, and e is in volts. 



dq de 



= l§i,<it. 



(b) For several condensers in parallel, the ' 
equivalent total capacitance is 

Ce^. = Ci -h Co -h Ca + • • • . 

(c) For several condensers in series, the 
equivalent total capacitance is given by the 
equation 



132 ELECTRICAL ENGINEERING 



^=1+1+1+ .... 

Cgq Ci Co Cz 

(d) Capacitance of a parallel-plate condenser. 

C = 0.08842 ^ 4 X 10-12 farads 
a 

= 0.08842 k— X 10-« microfarads, 
a 

where d is the uniform distance, in centimeters, 
between oppositely charged surfaces each of A 
square centimeters area, and k is the specific 
inductive capacity of the dielectric between. 
It is assumed that d is small in comparison 
with dimensions of plates. Values of k for 
common insulating materials are given on 
page 151. 

(e) Capacitance of single-conductor cable with 
grounded metal sheath. 

0.03882 A; V. .^^ r 

^ = 1 } — 1 — r X 10-« farads per mile, 

logio (Ti/ro) 

where Tq is the external radius of the inner 
cylindrical conductor and r^ is the internal 
radius of the outer sheath, both in terms of 
same units. Total capacitance is directly pro- 
portional to length, since capacitances of suc- 
cessive miles are all in parallel. 

(/) Capacitance to Neutral of each wire of a 
transmission line in air. 

0.03882 ,, ,^_«. , ., 

Co = \ , , , , X 10-6 farads per mile, 

logio (d/r) 

where r is the radius of the w^ire and d is the 
distance between centers, both in terms of 
same units. It is assumed that d is large com- 
pared with r, and both small compared with 
distance to surrounding objects. For a two- 
wire line, capacitance between wires (per mile 
distance) is one-half the value given above, 
as the condensers from each wire to neutral 



ELECTRICAL ENGINEERING 133 

are in series. For three wires spaced equi- 
distant (at vertices of equilateral triangle) as 
for three-phase line, the same formula gives 
capacitance to neutral per mile. 

(g) Charging Current per mile of a trans' 
mission line in air. 

If. = 2 tt/CoEq, 
where Eq equals 0.50 times r.m.s. value of 
volts between line wires for a single-phase 
two-wire line, and 0.577 times r.m.s. volts 
between wires for a three-phase three-wire 
line. Harmonic e.m.f. and balanced voltages 
are assumed. Iq is r.m.s. amperes if Eq is 
r.m.s. volts. Tables of charging current and 
line capacitance found in Handbooks are in 
accord with these formulae. 

(h) Energy stored in a condenser or in its 
dielectric. 



W, = I Cede = lce^ = l ' f = | Q^ 



-S' 



where Wc is the watt-seconds or joules required 
to raise condenser of C farads to a potential 
difference of e volts between terminals or 
plates, the charge being Q coulombs or ampere- 
seconds. 

CIRCUITS CARRYING DIRECT CURRENT 
(UNVARYING). 



4= 



a 

Fig. 7. 

When a conductor carries an unvarying cur- 
rent, the e.m.f. between any two points a and b 
is directly and exactly proportional to the 
current. That is, 

E 

—,= R = a constant 

?= resistance of conductor ah. 



CO w 

< "i 
•- -J 



134 ELECTRICAL ENGINEERING 

If £J is in volts and / in amperes, R is expressed 
as "ohms resistance." It is assumed that no 
e.m.f. is generated (as by battery or dynamo) 
between a and h. 

(b) Resistance of a conductor. 72 is a constant 
for any given temperature, material and dimen- 
sions of conductor; it varies with each of these 
factors as indicated in the following equations. 

R = Rq (1 -\- aQt) when dimensions and mate- 
rial of conductor remain unchanged. 

^0 = Po T when temperature is constant at 

Rq is ohms resistance at 0° C, and R is ohms 
for same conductor at t° C. uq, the tempera- 
ture coefficient for resistance, equals 0.00427 
for standard annealed copper and has practi- 
cally the same value for most pure metals 
(including aluminum, and soft steel) although 
it varies greatly among alloys, non-metals and 
solutions. See an Electrical Engineering 
Handbook, p is the resistivity, varying 
greatly with the nature and treatment of the 
conductor material; see page 152. I is the 
length of conductor and A its cross-sect i.-^'? 
area in plane normal to direction of current 
flow, in same units used to determine pq. 

^ . ^ ^ / length in feet \ 

Ro = 6.0153 po 1061 — ^_ . 

" V section area m circular-ii-ils/ 

ohms. 
One circular-mil is area of circle 0.001 inch 
diameter. 

For round, copper wires, at 20° C. or 08° F., 
the following relations form the basis for 
tables in Roebling's book "Wire in Electrical 

Construction ": 

10371 2 
Ohms per 1000 feet = ' ' • 

Pounds per 1000 feet = 0.003027 dK 



ELECTRICAL ENGINEERING 135 



d^ = section area in circ. mils 
= (diam. in inches X 1000)2. 

Values of d for standard (Brown & Sharpe) 
gauge numbers are given on page 153. 
(c) Total resistance of a series circuit. 



< 



J<-R.„olim3J 



VWW\Mr- 



-R-g-olims- 



Fig. 8. 

Rg = Ri -\- R2 -{■ R3. 
(d) Equivalent Resistance of a Parallel Circuit. 



Ri ohmg^ 




assuming that none of the paths contains any 
source of e.m.f. 

(e) Power lost in a conductor. 



P^ =E^I =IR XI = I^R = 



R 



where Pj. is the watts transformed into heat in 
a conductor of R ohms resistance carrying / 
amperes. E^. = IR is the volts consumed in 
overcoming resistance. 

(/) Series circuits carrying direct current. 
Relations of current, e.m.f., and power. 
_, I amperes R,ohmg 

~T" O 1 > 'VV\AV*^/V T— 



/ 








^' 




y^E^volts 


2 T 


— RgohmB 


/ 






I am 


iperes R^ oliin 
Fig. 10. 


s 





136 ELECTRICAL ENGINEERING 

Consider a generator G impressing an un- 
varying e.m.f. Ei upon a series circuit con- 
sisting of Ri -h Rz ohms, and a battery (or a 
motor) having internal resistance R^ ohms and 
a generated "back e.m.f." Eq volts directed 
opposite to Et (as indicated by dotted arrow). 
Then, if / represent the amperes flowing, 

Ef — Eq = I (Ri + -^2 + Ri). 
Et = El -\- E2 -{- E^ 

= IRi + iEo + IR.) + IR„ 

Pq = power output of generator = E^I watts. 

P^ = power transformed into heat in entire 
external circuit. 

P^ = P (Ri +R2+R3) watts. 

Px = power transformed chemically in bat- 
tery (or mechanically in motor) 
generating Eq. 

Py = EqI watts. 

P2 = power input to battery (or motor). 

P2 -= (^0 + IR2) I =EoI + PR2. 

If the connections are changed so that ^0 
acts in same direction as Ep then the sign of 
Eq is reversed in the above equations: 

Et+Eo=I (Ri + R2 + Rz)- 

P2 = - (^0 - ^^2) / = (^0/ - /2K2) 

watts output from generator of Eq. 

(g) Parallel circuits carrying direct current. 
Relations of current, e.m.f., and power. 



Fig I I , 




,<f^0 



ELECTRICAL ENGINEERING 137 

Consider a generator G impressing an un- 
varying e.m.f. Ei upon a load having resist- 
ance R^ ohms and internal e.m.f. E^ volts, 
over line wires having resistances Ri and R^ 
ohms; let a battery whose generated e.m.f. 
(on open-circuit) is Eq volts and internal 
resistance Rq ohms be connected in parallel 
with G to this same load R^. Directions of 
Eq and E^ are indicated by dotted arrows. 

Mark, as indicated, the directions in which 
currents 7i, 1 2, 1 5 in various parts of the circuit 
may flow; if the wrong direction happens to 
be chosen for any current, the algebraic solu- 
tion of the following simiiltaneous equations 
will give negative value for that current. We 
may now write: 

Ei — E^ = IiRi + I5R5 + I1R4 
or Et = E,+ I,R, = Et-h {Ri + ^4). 

and ^0 — -^5 ~ I2R2 4" -^2^0 ~h I2RZ ~f~ -^5^5 
or E-j = (Eq — liRo) — I1 {R2 + Rz) 

= Eq — /o (jK2 + R3) 
and Ii + I2 = If,' 

Numerical values having been assigned to 
Ei, Eq and E^ in volts, and to all the resistances 
in ohms, we should be able to find correspond- 
" ing values for 7i, J2, Iz by solving these equa- 
tions. 

{h) Solution of Networks. As indicated in 
the preceding examples, the solution of any 
series-parallel arrangement of circuits, or net- 
work, depends on the application of two 
principles, commonly known as Kirchoff's 
Laws: 

(a) In any closed circuit the algebraic sum 
of the products of the current and resistance 
in each of the conductors iji the circuit is equal 
to the electromotive force in the circuit. In 
applying this, account must be taken of the 



L 



138 ELECTRICAL ENGINEERING 

relative direction of the e.m.f.'s and the cur- 
rents in various parts of the circuit. 

(6) The algebraic sum of the currents which 
meet at any point is zero; or, the sum of 
currents toward a juncture must be equal to 
the sum of currents away from that juncture. 

DIRECT-CURRENT MACHINES. 

(a) Electromotive force generated in the arma- 
ture between terminals is 

where the armature has altogether Z conductors 
on its outer surface arranged in s parallel cir- 
cuits, and revolves at n revolutions per second 
in a field of p poles from each of which ^ max- 
wells enter the armature. If the dynamo 
operates as generator / is in same direction as 
Eg-, if it operates as motor, Eg is in opposition 
to /. Brushes are assumed to be on neutral 
points. 

(6) Terminal voltage of a d-c. dynamo is 

Ef = Eg zh {Ra^a + ^se^se + Rcp^cp)* 

where Ra, Rge, Re are the resistances of arma- 
ture, series field and commutating-poles, re- 
spectively, in ohms; and /«, Iger Icp ^-re the 
currents in the corresponding parts, amperes. 
The + sign is used if the dynamo operates as 
motor, the — sign if it operates as generator. 
For shunt-wound dynamo the Rgelae term is 
omitted, and if it has no commutating-poles 
omit the term R^pl ep- 
ic) Torque of a dynamo is 

^ 0.1174 p$Z7„ ,, ^ 

T = rrrf pound-feet, 

II/* s 

where T is the total torque magnetically de- 
veloped on an armature with Z surface conduc- 



ELECTRICAL ENGINEERING 139 

tors arranged in s parallel paths, due to a total 
current /^ amperes, when $ maxwells enter the 
armature from each of p poles. The torque at 
pulley must be slightly greater than this in a 
generator, or slightly less in a motor, on account 
of friction (and magnetic losses if the dynamo 
is rotating). 

{d) Speed of a motor is 

108 Egs 108s {Et - RI) 

^=-^^z" = ^¥z — -^^^-P^^^^-' 

where RI is the total resistance drop in the 
armature circuit across which E^ volts is im- 
pressed, including series field and commutating- 
pole winding if the motor has such. 

(e) Efficiency and Losses in a d-c. dynamo. 

For a Generator: 

_ watts output _ EfTt 

"^ ~ watts input ~ EJi + Pf+ P,, + P^ + P/ 

For a Motor: 

Pj, = Ij^Rj> = heat loss in shunt field coils and 
rheostat. 

Pae = lae^se = li^at loss in series field coils and 
regulating shunt. 

Pep = Icp^cp = lieat loss in commutating-pole 
winding. 

Pg = stray power, including hysteresis and 
eddy-current losses in armature core 
and in pole faces, and friction losses in 
bearings, brushes and windage. 

GROWTH AND DECAY OF CURRENT 
IN INDUCTIVE CIRCUIT. 

where i is the amperes flowing in a circuit having 
resistance R ohms and self-inductance L henrys 



00 w 

< ^ 



140 ELECTRICAL ENGINEERING 

arranged in series, at an instant t seconds after 
an unvarying e.m.f. E volts has been applied. 
Current assumed to start from zero. 

If the impressed e.m.f. E is removed from a 
circuit of resistance R ohms and self-inductance 
L henrys, when it is carrying a steady current 

E 

I = —, and the circuit is closed through an 
K 

additional resistance Ri ohms, the current be- 
comes i amperes at an instant t seconds after, 

where 

_ B + Rx ^ ^ __ B + Ri ^ 
% = le L = o e L 

The amount of e.m.f. generated in the coil at 
this instant is 

e =L—=i{R-\-Ry) =E e L 

General Equation for electric circuit having 
resistance R ohms, self-inductance L henrys, 
and capacitance C farads, all in series is 



/•■■ 



. '.dt 
at u 

where e volts applied produces a current i 
amperes which is changing at the rate — am- 
peres per second. This relation holds at every 
instant, for any mode of variation of e.m.f. or 
current. 



HARMONIC ALTERNATING CURRENT. 

A simple harmonic e.m.f. which completes / 
cycles per second has a value e volts at an in- 
stant t seconds after it has attained its maximum 
positive value E^ volts, where 

e = E„^ cos 2Trft = E^^ cos uit. 



ELECTRICAL ENGINEERING 141 

This e.m.f. will produce a simple harmonic 
current (i amperes) in any circuit having re- 
sistance R ohms, self -inductance L henrys, and 
3apacitance C farads, where 

I = Ijn cos ((x}t — 6), 

E 



I.. = 



Sl^'+(-^-Zcf 



Em Er 



9 = arc tan p = arc cos 



(i) 



Effective or square-root-mean-square value of 
this e.m.f. is 

E =^ = 0.101 Em, 

and of this current is 

I =0.7071^ =|- 



Average 


value of this e.m.f. (during 


one 


half- 


cycle) is 




--Em- 










Eav 


0.636 Em. 






Form-factor = 


Eeffective 
" average 


0.707 
0.636 








= 


1.11 for 
current 


harmonic 


e.m 


f. Ol 



E 



Impedance = Z = j= ^R^ + X\ 

Reactance = X = ( 2 irfL — - — — :; J • 

Power at any instant in a circuit where E 
(r.m.s.) volts produces / (r.m.s.) amperes, lag- 
ging (or leading) 6 electrical degrees i or ;—-: of 

\ ooU 

- seconds ) with respect to £^, is 






142 ELECTRICAL ENGINEERING 

V (watts) = ie = EI cos d + EI cos (4 irft — d) 

= E^ cos cot X Im COS (cot — 6). 

Average power in this circuit is 
P = EI cos 6 = average of p for complete cycle. 
P 



Power-factor = 



power 



EI apparent power 
= cos d (when e and i are har- 



IZ'I 



Series Circuits carrying simple harmoni: 
Alternating Current. 




c-^ 



Fig. 12. 
R = Ri -j- R2 ~1~ ^3« 



rij, lagging 



O 

-El 



n 21 lagging 



o 

-E7 



rio pleading 



o 



Fig. 13, 



Three units in series have e.m.f.'s Eu Eo, Ei 
(r.m.s. volts) respectively, power factors yni, 
mn, mz, ancj reactive factors ni, n2, ^3 respec- 
tively, where 

m = cos 6 

n = sind = ^1 — m2. 

All carry the same current, I (r.m.s. amperes). 



ELECTRICAL ENGINEERING 143 



Component of 
'Q~ ^ E in phase with / 
<^ = miEi + moE-z + 

m^Ei = Er. 

Component of 
E at 90° to 7 =^ 
►^ n,Ei + n2E2 + 
rizEz = Ex- 

If I lags, nE is 
positive ; if I leads E, then nE is negative. 




Fig. 14. 



Total voltage = E = ^Ej^ + E^^. 



Total power factor — m = cos d - 
Total reactive factor = n — sin 



Er 

E ' 
Ex 

E ' 



Total power = P = P^ + P^ + P^ = miEJ 
+ m^E^I + mzEzT = mEI. 

Total reactive volt-amperes = niLiI -{- 
n^EiI + nzEzI = nEI. 

Total apparent power = ^(mEI)^ + {nEiy^ 
= EI (volt-amperes). 

Parallel Circuits carrying simple harmonio 
Alternating Current. 





^^^^ — 'WJW — 

II 


X,2 






I 


h 


■ I 








h 















Z2=\/R22+X22 

Fig 15 




Fig. 16. 



144 ELECTRICAL ENGINEERING 
Component of h in phase with E equals 

Component of I2 in phase with E equals 

Component of I in phase with E equals 

(^1 +02)E = gE. 
Component of Ii at 90° to J^ equals 

Component of I2 at 90° to E equals 
^^x| = ixi?=6,B. 

Component of / at 90" to E equals 

(61 + 62) E = bE. 

When the current leads, b is considered as 
positive, and when the current lags b is negative, 

/ = V(^^)2 4. (bE)^ = E V^2 + 62 = yE. 

Equivalent impedance 



/ y Vg2 -\- 62 

Equivalent resistance 

y g^ + b^ 
Equivalent reactance 

Instead of a simple combination of R, L and 
C, path No. 1 may be an induction motor taking 

Ii amperes at E volts with power factor mi= -^ t 

reactive factor ni = -^r (lagging); while path 

No. 2 may be an over-excited synchronous 
motor taking 1 2 amperes at E volts with power 



ELECTRICAL ENGINEERING 145 

factor m^ — it ^ reactive factor 712=-^ (lead- 

ing). In this case 61 and 62 would have opposite 
signs but inasmuch as both paths take in power, 
g will have the same sign in both cases. 
Total power =P=Pi +P2 =Ii^Ri -\-UR'^ =PReq. 

= miEIi -\-m2Eh =9iE^ +92E^ 

=mEI=gEK 

Total reactive volt-amperes 

= /i2Xi + 72^X2 = I^Xeq. 

= n^EIx + n.El2 = 61^2 4. 5,^2 

= nEI = hE^. 

Total apparent power = ^{mEI)'^ + (n^/)2 

= £?2 V^2 _|_ ^2 

= EI (volt-amperes). 
mill -\- mill 



Total power factor 



gE g 

m = 



I Vg2 4. 52 

Conductance, g, mhos. 
Susceptance, h, mhos. 
Admittance, y, mhos. 

The significance and use of these three quanti- 
ties, and the relation of each to R and X in either 
geries or parallel circuits, should be evident from 
the preceding examples. 

THREE-PHASE CIRCUITS. 

(a) Star or Wye Connection. 
A^ ^ 




F^g. 17, 



146 ELECTRICAL ENGINEERING 




Fig. 18. 

For balanced or unbalanced condition: 

Note. — The dots indicate that vectors, not 
arithmetic values, are added. 

?AB = ?AN + ?NB = ~ ?NA + ^NB- 
?BC = ?BN + ?NC = ~ ?NB + ^NC* 
?CA = ?CN + ?NA = ~ ?NC + ^NA- 



NA, 



^B - ^NB' 



^C - ^NC- 



Jna +^ 



NB '^ InC 



(if no current flows in a neutral connection). 



E 



AB 



+ ^Ttn + ^rA = 



'BC 



•^CA 



Total power = E^^I^^ cos 6^^ + E^^Jnb 
cos d^j^ + JB^^c^i^C co-'^ ^iVC 
The three phases are "balanced" when 



Ja 


= 


Ib = 


^C 


^AB 


= 


^BC 


= ECA 


^NA 


= 


^NB 


= %c. 



VsE 



NA- 



NA 



NB 



'NC- 



Phase angle between line voltage and phase 
voltage is 30° when phases are balanced. 



ELECTRICAL ENGINEERING 147 



Delta or Mesh Connection. 
t^^ h 




Fig. 19. 




Fig. 20. 
For balanced or unbalanced condition: 



I A = ICA + Iba 


^ ^CA ~ ^.AB- 


Ib "" \aB + f CJB 


= {^J5 "" ^.BC- 


Ic = IbC + ^.AC 


= ^.BC ~ ^CA' 



\ (For simplicity, only the first of these equations 
is illustrated in Fig. 20.) 



?AB + ?BC + ?CA - ^' 
^AB^AB COS e^^ 



^BC^BC 



Total power 

When the three phases are balanced, we have 



< "J 
h -J 



148 ELECTRICAL ENGINEERING 



^A=^B= ^C 



Vll 



AB- 



AB 



~ ^BC - ^CA- 



^AB - 



^BQ - ^CA- 



Phase angle between line current and ph£ 
current is 30° when phases are balanced, 
(c) Power in Three-phase Systems. 



il 



Po 



Fig. 21. 

Let Ep, I p and cos Op be the e.m.f., current 
and power factor within each of the three 
phases, which may be connected either in wye 
or in delta; let E j^ be the e.m.f. between line 
wires, I j^ the current in each line wire, and cos 6j^ 
the power factor of the entire system. Then, 
for balanced system, 

Total power = 3 Epjp cos dp = \/^ ^l^l ^^^ ^L 

total watts 

"V 



^ total volt-amperes 



= cos 



In the balanced system, the total power is 
unvarying — is the same at every instant. 

Fig. 21 shows how to connect two identically 
similar wattmeters (Pi and P?) so that the alge- 



ELECTRICAL ENGINEERING 149 

braic sum of their indications equals the total 
power being transmitted over any three-wire 
system ABC (which may be three-phase). 
This is correct for any power factor, and for 
either balanced or unbalanced loads. For 
balanced loads the values Pi and P2 are equal 
at power factor 1.00; one of them becomes zero 
at power factor 0.50, and becomes negative for 
power factors lower than 0.50. 

Power factor may be calculated from the 
wattmeter readings if load is balanced, as follows: 

■Pi + P2 
cos 



2 VPj2 - P1P2 + Pz^ 

TRANSFORMERS: VOLTAGE AND 
CURRENT RATIOS 

If practically all flux links both primary and 
secondary coils, as is usually the case in 
" constant voltage transformers," the ratio of 
primary turns in series to secondary turns in 
series is equal to the ratio of e.m.f. between 
primary terminals to e.m.f. between secondary 
terminals at zero load, or is equal to the in- 
verse of the ratio of primary load current to 
secondary load current. 

TRANSFORMERS: VOLTAGE 
REGULATION 

With low-tension coils short-circuited, meas- 
ure the *' impedance volts " {E^ = ZI) necessary 

\ to impress upon high-tension winding to pro- 
duce full-load current 7^ in high-tension circuit, 

and measure also the (total) " impedance watts" 
PgQ then being supplied to the transformer. 
Then 

— — ^^^— ^— ^C 

{XI) = VEy2 - {RI)\ and (RI) = -1-- 



150 ELECTEICAL ENGINEERING 



from which we draw the following diagram as 
for a simple series circuit: 

E = 100% 




Fig. 22. 

wherein E' { = 100% of itself) represents rated 
high-tension voltage, cos d is the power factor of 
load between secondary terminals, {RI%) and 
(XI%) designate the e.m.f.'s RI and XI re- 
ferred to above but expressed now as percent- 
ages of the rated high-tension voltage. From 
this diagram it follows that (Ei%)^ = 
(100 cos d + i2/%)2 -{- (100 Vi_(cos0)2 +XJ%)2 

Per cent voltage regulation of transformer = 
Ei%— 100, wherein (Ei %) represents e.m.f . 
necessary to impress upon high-tension coils, as 
per cent of rated h-t e.m.f. 

Transformers in parallel are treated as imped- 
ances in parallel, since E' must be the same for 
all that are paralleled, as also Ei and ZI. 



DISTRIBUTING LINES AND SHORT 
TRANSMISSIONS : REGULATION 

The same diagram and equations given above 
for transformers should serve also for calculating 
voltage regulation of short transmissions (where 
the distribution of capacitance, inductance and 
leakance need not be considered). In this case 
E' represents the voltage required to be deliv- 
ered at load end, 7^ is total current in line to 

load, {RI%) is the resistance drop as percentage 
of load voltage E', (XI%) is the reactance drop 



ELECTRICAL ENGINEERING 151 

as per cent, cos d is the power factor of the load, 
(Ei%) is the e.m.f . necessary to impress at input 
end of line as per cent of load voltage. At zero 
load E' changes so as to equal Eu therefore the 
change of load voltage is the algebraic difference 
between E' and Ei. 

Wire Table for Round Wires. 



Gauge 




Guage 




Number, 


Diameter 


Number, 


Diameter 


Brown & 


in Mils. 


Brown & 


In Mils. 


Sharpe. 




Sharpe. 




0000 


460.0 


19 


36.0 


COO 


410.0 


20 


32.0 


00 


365.0 


21 


28.5 





325.0 


22 


25.3 


1 


289.0 


23 


22.6 


2 


258.0 


24 


20.1 


3 


229.0 


25 


17.9 


4 


204.0 


26 


15.9 


5 


182.0 


27 


14.2 


6 


162.0 


28 


12.6 


7 


144.0 


29 


11.3 


8 


128.0 


30 


10.0 


9 


114.0 


31 


8.9 


10 


102.0 


32 


8.0 


11 


91.0 


33 


7.1 


12 


81.0 


34 


6.3 


13 


72.0 


35 


5.6 


14 


64.0 


36 


5.0 


15 


57.0 


37 


4.5 


16 


51.0 


38 


4.0 


17 


45.0 


39 


3.5 


18 


40.0 


40 


3.1 



i mil = 0.001 inch. 



pw 



152 ELECTRICAL ENGINEERING 



Magnetization Curves for Electrical Steels. 





Ampere-tiirns pef Inch Length 


Kilo-Max- 


of Magnetic Circuit. 


wells per 
Square Inch. 












Sheets. 


Castings. 


10 
20 
30 


1.32 
1.66 
2.03 






8.82 


40 


2.48 


11.4 


50 


3.07 


14.5 


60 


3.97 


18.5 


70 


5.50 


24.3 


80 


8.30 


36.0 


90 


15.30 


56.7 


100 


40.00 


97.0 


110 


135.00 


182.00 


120 


336.00 


370.00 


130 


1050.00 


1100.00 



Data from Pender's Handbook for Electrical 
Engineers. 

Hysteresis Loss, Watts per Pound, at 60 Cycles 
when B„, = 10,000 Gausses. . 



Metal. 


Range of Values. 


From 


To 


Silicon Steel Annealed 
Sheets 

Ordinary Electrical Sheets, 
Annealed 

Soft Cast Steel 


0.55 

0.84 

2.7 

10.00 

13.00 


1.36 

3.5 
11.00 
14.00 
22.00 


Cast Iron 


Forged Steel 





L 



Specific gravity : 

Ordinary Electrical Sheets = 7.7. 

Silicon Steel = 7.5. 
Data from Pender's Handbook for Electrical 
Engineers. 



ELECTRICAL ENGINEERING 153 



Eddy-current Loss, Watts per Pound, for Sheets 
t 0.0141 Inch Thick, at 60 Cycles 

when Brr, = 10,000 Gausses. 



Kind of Sheets. 


Range of Values. 


From 


To 


Average. 


Silicon Steel 


0.12 
0.34 


0.27 
0.70 


0.18 
0.608 


Ordinary Electrical. . 



Data from Pender's Handbook for Electrical 
Engineers. 



Dielectric Constants. 



Substance. 



i 



Air 

Glass 

Rubber 

I 

; Gutta Percha. 

Mica 

Paper 

Oil 

Paraffin 

Shellac 



Value of Jc. 



1.00 
5.5 to 10.0 
2.0to4.0 

2.9 
2.5 to 5.9 
1.7to4.0 
2.0to2.5 
1.9 to2.3 
2.7 to 3.8 



Data from Standard Handbook for Electrical 
Engineers. 



P-i 



154 ELECTRICAL ENGINEERING 



















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TABLES 



I. LOGARITHMS OF NUMBERS 

II. LOGARITHMIC SINES AND CO- 
SINES 

III. LOGARITHMIC TANGENTS AND 

COTANGENTS 

IV. NATURAL SINES AND COSINES 

V. NATURAL TANGENTS AND CO- 
TANGENTS 
VI. CONVERSION FACTORS 

VII. PROPERTIES OF SATURATED 
STEAM 

VIII. PRESSURE-ENTROPY TABLE FOR 
STEAM 



195 



h -J 



I. LOGARITHMS 



L 



N 





1 


2 


3 


4 


D 


100 


00000 


00043 


00087 


00130 


00173 


43 


1 


0432 


0475 


0518 


0561 


0604 


43 


2 


0860 


0903 


0945 


0988 


1030 


42 


3 


1284 


1326 


1368 


1410 


1452 


42 


4 


1703 


1745 


1787 


1828 


1870 


42 


5 


02119 


02160 


02202 


02243 


02284 


41 


6 


2531 


2572 


2612 


2653 


2694 


41 


7 


2938 


2979 


3019 


3060 


3100 


40 


8 


3342 


3383 


3423 


3463 


3503 


40 


9 


3743 


3782 


3822 


3862 


3902 


40 


110 


04139 


04179 


04218 


04258 


04297 


39 


1 


4532 


4571 


4610 


4650 


4689 


39 


2 


4922 


4961 


4999 


.5038 


5077 


39 


3 


5308 


5346 


5385 


5423 


5461 


38 


4 


5690 


5729 


5767 


5805 


5843 


38 


5 


06070 


06108 


06145 


06183 


06221 


38 


6 


6446 


6483 


6521 


6558 


6595 


37 


7 


6819 


6856 


6893 


6930 


6967 


37 


8 


7188 


7225 


7262 


7298 


7335 


37 


9 


7555 


7591 


7628 


7664 


7700 


36 


120 


07918 


07954 


07990 


08027 


08063 


36 


1 


8279 


8314 


8350 


8386 


8422 


36 


2 


8636 


8672 


8707 


8743 


8778 


35 


3 


8991 


9026 


9061 


9096 


9132 


35 


4 


9342 


9377 


9412 


9447 


9482 


35 


6 


09691 


09726 


09760 


09795 


09830 


35 


6 


10037 


10072 


10106 


10140 


10175 


34 


7 


0380 


0415 


0449 


0483 


0517 


34 


8 


0721 


0755 


0789 


0823 


0857 


34 


9 


1059 


1093 


1126 


1160 


1193 


33 


130 


11394 


11428 


11461 


11494 


11528 


33 


1 


1727 


1760 


1793 


1826 


1860 


33 


2 


2057 


2090 


2123 


2156 


2189 


33 


3 


2385 


2418 


2450 


2483 


2516 


32 


4 


2710 


2743 


2775 


2808- 


2840 


32 


6 


13033 


13066 


13098 


13130 


13162 


32 


6 


3354 


3386 


3418 


3450 


3481 


32 


7 


3672 


3704 


3735 


3767 


3799 


32 


8 


3988 


4019 


4051 


4082 


4114 


31 


9 


4301 


4333 


4364 


4395 


4426 


31 


140 


14613 


14644 


14675 


14706 


14737 


31 


1 


4922 


4953 


4983 


5014 


5045 


31 


2 


5229 


5259 


5290 


5320 


5351 


30 


3 


5534 


5564 


5594 


5625 


5655 


30 


4 


5836 


5866 


5897 


5927 


5957 


30 


5 


16137 


16167 


16197 


16227 


16256 


30 


6 


6435 


6465 


6495 


6524 


6554 


30 


7 


6732 


6761 


6791 


6820 


6850 


29 


8 


7026 


7056 


7085 


7114 


7143 


29 


9 


7319 


7348 


7377 


7406 


7435 


29 


150 


17609 


17638 


17667 


17696 


17725 


29 


1 


7898 


7926 


7955 


7984 


8013 


29 


2 


8184 


8213 


8241 


8270 


8298 


28 


3 


8469 


8498 


8526 


8554 


8583 


28 


4 


8752 


8780 


8808 


8837 


8865 


28 


6 


19033 


19061 


19089 


19117 


19145 


28 


6 


9312 


9340 


9368 


9396 


9424 


28 


7 


9590 


9618 


9645 


9673 


9700 


28 


8 


9866 


9893 


9921 


9948 


9976 


27 


9 


20140 


20167 


20194 


20222 


20249 


27 



156 



OF NUMBERS 



N 


5 


6 


7 


8 


9 


D 


100 


00217 


00260 


00303 


00346 


00389 


43 


1 


0647 


0689 


0732 


0775 


0817 


43 


3 


1072 


1115 


1157 


1199 


1242 


42 


3 


1494 


1536 


1578 


1620 


1662 


42 


4 


1912 


1953 


1995 


2036 


2078 


42 


5 


02325 


02366 


02407 


02449 


02490 


41 


6 


2735 


2776 


2816 


2857 


2898 


41 


7 


3141 


3181 


3222 


3262 


3302 


40 


8 


3543 


3583 


3623 


3663 


3703 


40 


9 


3941 


3981 


4021 


4060 


4100 


40 


110 


04336 


04376 


04415 


04454 


04493 


39 


1 


4727 


4766 


4805 


4844 


4883 


39 


2 


5115 


5154 


5192 


5231 


5269 


39 


3 


5500 


5538 


5576 


5614 


5652 


38 


4 


5881 


5918 


5956 


5994 


6032 


3S 


5 


06258 


06296 


06333 


06371 


06408 


38 


6 


6633 


6670 


6707 


6744 


6781 


37 


7 


7004 


7041 


7078 


7115 


7151 


37 


8 


7372 


7408 


7445 


7482 


7518 


37 


9 


7737 


7773 


7809 


7846 


7882 


36 


120 


08099 


08135 


08171 


08207 


08243 


36 


1 


8458 


8493 


8529 


8565 


8600 


36 


2 


8814 


8849 


8884 


8920 


8955 


35 


3 


9167 


9202 


9237 


9272 


9307 


35 


4 


9517 


9552 


9587 


9621 


9656 


35 


5 


09864 


09899 


09934 


09968 


10003 


35 


6 


10209 


10243 


10278 


10312 


0346 


34 


7 


0551 


0585 


0619 


0653 


0687 


34 


8 


0890 


0924 


0958 


0992 


1025 


34 


9 


1227 


1261 


1294 


1327 


1361 


33 


130 


11561 


11594 


11628 


11661 


11694 


33 


1 


1893 


1926 


1959 


1992 


2024 


33 


2 


2222 


2254 


2287 


2320 


2352 


33 


3 


2548 


2581 


2613 


2646 


2678 


32 


4 


2872 


2905 


2937 


2969 


3001 


32 


6 


13194 


13226 


13258 


13290 


13322 


32 


6 


3513 


3545 


3577 


3609 


3640 


32 


7 


3830 


3862 


3893 


3925 


3956 


32 


8 


4145 


4176 


4208 


4239 


4270 


31 


9 


4457 


4489 


4520 


4551 


4582 


31 


140 


14768 


14799 


14829 


14860 


14891 


31 


1 


5076 


5106 


5137 


5168 


5198 


31 


2 


5381 


5412 


5442 


5473 


5503 


30 


3 


5685 


5715 


5746 


5776 


5806 


30 


4 


5987 


6017 


6047 


6077 


6107 


30 


5 


16286 


16316 


16346 


16376 


16406 


30 


6 


6584 


6613 


6643 


6673 


6702 


30 


7 


6879 


6909 


6938 


6967 


6997 


29 


8 


7173 


7202 


7231 


7260 


7289 


29 


9 


7464 


7493 


7522 


7551 


7580 


29 


150 


17754 


17782 


17811 


17840 


17869 


29 


1 


8041 


8070 


8099 


8127 


8156 


29 


2 


8327 


8355 


8384 


8412 


8441 


28 


3 


8611 


8639 


8667 


8696 


8724 


28 


4 


8893 


8921 


8949 


8977 


9005 


28 


5 


19173 


19201 


19229 


19257 


19285 


28 


6 


9451 


0479 


9507 


9535 


9562 


28 


7 


9728 


9756 


9783 


9811 


9838 


28 


8 


20003 


20030 


20058 


20085 


20112 


27 


9 


0276 


0303 


0330 


0358 


0385 


27 



157 



I. LOGARITHMS 



L 



N 





1 


2 


3 


4 


D 


160 


20412 


20439 


20466 


20493 


20520 


27 


1 


0683 


0710 


0737 


0763 


0790 


27 


2 


0952 


0978 


1005 


1032 


1059 


27 


3 


1219 


1245 


1272 


1299 


1325 




4 


1484 


1511 


1537 


1564 


1590 


26 


6 


21748 


21775 


21801 


21827 


21854 


26 


6 


2011 


2037 


2063 


2089 


2115 


26 


7 


2272 


2298 


2324 


2350 


2376 


26 


8 


2531 


2557 


2583 


2608 


2634 


26 


9 


2789 


2814 


2840 


2866 


2891 


26 


170 


23045 


23070 


23096 


23121 


23147 


25 


1 


3300 


3325 


3350 


3376 


3401 


25 


2 


3553 


3578 


3603 


3629 


3654 


25 


3 


3805 


3830 


3855 


3880 


3905 


25 


4 


4055 


4080 


4105 


4130 


4155 


25 


5 


24304 


24329 


24353 


24378 


24403 


25 


6 


4551 


4576 


4601 


4625 


4650 


25 


7 


4797 


4822 


4846 


4871 


4895 


24 


8 


5042 


5066 


5091 


5115 


5139 


24 


9 


5285 


5310 


5334 


5358 


5382 


24 


180 


25527 


25551 


25575 


25600 


25624 


24 


1 


5768 


5792 


5816 


5840 


5864 


24 


2 


6007 


6031 


6055 


6079 


6102 


24 


3 


6245 


6269 


6293 


6316 


6340 


24 


4 


6482 


6505 


6529 


6553 


6576 


24 


5 


26717 


26741 


26764 


26788 


26811 


23 


6 


6951 


6975 


6998 


7021 


7045 


23 


7 


7184 


7207 


7231 


7254 


7277 


23 


8 


7416 


7439 


7462 


7485 


7508 


23 


9 


7646 


7669 


7692 


7715 


7738 


23 


190 


27875 


27898 


27921 


27944 


27967 


23 


1 


8103 


8126 


8149 


8171 


8194 


23 


2 


8330 


8353 


8375 


8398 


8421 


23 


3 


8556 


8578 


8601 


8623 


8646 


22 


4 


8780 


8803 


8825 


8847 


8870 


22 


5 


29003 


29026 


29048 


29070 


29092 


22 


6 


9226 


9248 


9270 


9292 


9314 


22 


7 


9447 


9469 


9491 


9513 


9535 


22 


8 


9667 


9688 


9710 


.9732 


9754 


22 


9 


9885 


9907 


9929 


9951 


9973 


22 


200 


30103 


30125 


30146 


30168 


30190 


22 


1 


0320 


0341 


0363 


0384 


0406 


22 


2 


0535 


0557 


0578 


0600 


0521 


21 


3 


0750 


0771 


0792 


0814 


0835 


21 


4 


0963 


0984 


1006 


1027 


1048 


21 


6 


31175 


31197 


31218 


31239 


31260 


21 


6 


1387 


1408 


1429 


1450 


1471 


21 


7 


1597 


1618 


1639 


1660 


1681 


21 


8 


1806 


1827 


1848 


1869 


1890 


21 


9 


2015 


2035 


2056 


2077 


2098 


21 


210 


32222 


32243 


32263 


32284 


32305 


21 


1 


2428 


2449 


2469 


2490 


2510 


21 


2 


2634 


2654 


2675 


2695 


2715 


20 


3 


2838 


2858 


2879 


2899 


2919 


20 


4 


3041 


3062 


3082 


3102 


3122 


20 


5 


33244 


33264 


33284 


33304 


33325 


20 


6 


3445 


3465 


3486 


3506 


3526 


20 


7 


3646 


3666 


3686 


3706 


3726 


20 


8 


3846 


3866 


3885 


3905 


3925 


20 


9 


4044 


4064 


4084 


4104 


4124 


20 



158 



OF NUMBERS 



N 


5 


6 


7 


8 


9 


D 


160 


20548 


20575 


20602 


20629 


20656 


"27 


1 


0817 


0844 


0871 


0898 


0925 


27 


2 


1085 


1112 


1139 


1165 


1192 


27 


3 


1352 


1378 


1405 


1431 


1458 


26 


4 


1617 


1643 


1669 


1696 


1722 


26 


5 


21880 


21906 


21932 


21958 


21985 


26 


6 


2141 


2167 


2194 


2220 


2246 


26 




2401 


2427 


2453 


2479 


2505 


26 


g 


2660 


2686 


2712 


2737 


2763 


26 


9 


2917 


2943 


2968 


2994 


3019 


26 


170 


23172 


23198 


23223 


23249 


23274 


25 


1 


3426 


3452 


3477 


3502 


3528 


25 


3 


3679 


3704 


3729 


3754 


3779 


25 


3 


3930 


3955 


3980 


4005 


4030 


25 


4 


4180 


4204 


4229 


4254 


4279 


25 


5 


24428 


24452 


24477 


24502 


24527 


25 


6 


4674 


4699 


4724 


4748 


4773 


25 


7 


4920 


4944 


4969 


4993 


5018 


24 


8 


5164 


5188 


5212 


5237 


5261 


24 


9 


5406 


5431 


5455 


5479 


5503 


24 


180 


25648 


25672 


25696 


25720 


25744 


24 


1 


5888 


5912 


5935 


5959 


5983 


24 


2 


6126 


6150 


6174 


6198 


6221 


24 


3 


6364 


6387 


6411 


6435 


6458 


24 


4 


6600 


6623 


6647 


6670 


6694 


24 


5 


26834 


26858 


26881 


26905 


26928 


23 


6 


7068 


7091 


7114 


7138 


7161 


23 


7 


7300 


7323 


7346 


7370 


7393 


23 


8 


7531 


7554 


7577 


7600 


7623 


23 


9 


7761 


7784 


7807 


7830 


7852 


23 


190 


27989 


28012 


28035 


28058 


28081 


23 


1 


8217 


8240 


8262 


8285 


8307 


23 


2 


8443 


8466 


8488 


8511 


8533 


23 


3 


8668 


8691 


8713 


8735 


8758 


22 


4 


8892 


8914 


8937 


8959 


8981 


22 


6 


29115 


29137 


29159 


29181 


29203 


22 


6 


9336 


9358 


9380 


9403 


9425 


23 


7 


9557 


9579 


9601 


9623 


9645 


23 


8 


9776 


9798 


9820 


9842 


9863 


22 


9 


9994 


30016 


30038 


30060 


30081 


22 


200 


30211 


30233 


30255 


30276 


30298 


23 


1 


0428 


0449 


0471 


0492 


0514 


22 


2 


0643 


0664 


0685 


0707 


0728 


21 


3 


0856 


0878 


0899 


0920 


0942 


21 


4 


1069 


1091 


1112 


1133 


1154 


21 


5 


31281 


31302 


31323 


31345 


31366 


21 


6 


1492 


1513 


1534 


1555 


1576 


21 


7 


1702 


1723 


1744 


1765 


1785 


21 


8 


1911 


1931 


1952 


1973 


1994 


21 


9 


2118 


2139 


2160 


2181 


2201 


21 


210 


32325 


32346 


32366 


32387 


32408 


21 


1 


2531 


2552 


2572 


2593 


2613 


21 


2 


2736 


2756 


2777 


2797 


2818 


20 


i 3 


2940 


2960 


2980 


3001 


3021 


20 


4 


3143 


3163 


3183 


3203 


3224 


20 


. 5 


33345 


33365 


33385 


33405 


33425 


20 


6 


3546 


3566 


3586 


3606 


3626 


20 


7 


3746 


3766 


3786 


3806 


3826 


20 




3945 


3965 


3985 


4005 


4025 


20 


9 


1 4143 


4163 


4183 


4203 


4223 


20 



159 



I. LOGARITHMS 



N 





1 


2 


3 


4 


d' 


220 


34242 


34262 


34282 


34301 


34321 


20 


1 


4439 


4459 


4479 


4498 


4518 


20 


2 


4635 


4655 


4674 


4694 


4713 


20 
19 t 


3 


4830 


4850 


4869 


4889 


4908 


4 


5025 


5044 


5064 


5083 


5102 


19 


5 


35218 


35238 


35257 


35276 


35295 


19 


6 


5411 


5430 


5449 


5468 


5488 


19 


7 


5603 


5622 


5641 


5660 


5679 


19 


8 


5793 


5813 


5832 


5851 


5870 


19 


9 


5984 


6003 


6021 


6040 


6059 


19 


330 


36173 


36192 


36211 


36229 


36248 


19 


1 


6361 


6380 


6399 


6418 


6436 


19 


2 


6549 


6568 


6586 


6605 


6624 


19 


3 


6736 


6754 


6773 


6791 


6810 


19 


4 


6922 


6940 


6959 


6977 


6996 


18 


5 


37107 


37125 


37144 


37162 


37181 


18 


6 


7291 


7310 


7328 


7346 


7365 


18 


7 


7475 


7493 


7511 


7530 


7548 


18 


8 


7658 


7676 


7694 


7712 


7731 


18 


^ 9 


7840 


7858 


7870 


7894 


7912 


18 


240 


38021 


38039 


38057 


38075 


38093 


18 


1 


8202 


8220 


8238 


8256 


8274 


18 


2 


8382 


8399 


8417 


8435 


8453 


18 


3 


8561 


8578 


8596 


8614 


8632 


18 


4 


8739 


8757 


8775 


8792 


8810 


18 


5 


38917 


38934 


38952 


38970 


38987 


18 


6 


9094 


9111 


9129 


9146 


9164 


18 


7 


9270 


9287 


9305 


9322 


9340 


18 


8 


9445 


9463 


9480 


9498 


9515 


17 


9 


9620 


9637 


9655 


9672 


9690 


17 


250 


39794 


39811 


39829 


39846 


39863 


17 


1 


9967 


9985 


40002 


40019 


40037 


17 


2 


40140 


40157 


0175 


0192 


0209 


17 


3 


0312 


0329 


0346 


0364 


0381 


17 


4 


0483 


0500 


0518 


0535 


0552 


17 


5 


40654 


40671 


40688 


40705 


40722 


17 


6 


0824 


0841 


0858 


0875 


0892 


17 


7 


0993 


1010 


1027 


1044 


1061 


17 


8 


1162 


1179 


1196 • 


1212 


1229 


17 


9 


1330 


1347 


1363 


1380 


1397 


17 


260 


41497 


41514 


41531 


41547 


41564 


17 


1 


1664 


1681 


1697 


1714 


1731 


17 


2 


1830 


1847 


1863 


1880 


1896 


17 


3 


1996 


2012 


2029 


2045 


2062 


16 


4 


2160 


2177 


2193 


2210 


2226 


16 


5 


42325 


42341 


42357 


42374 


42390 


16 


6 


2488 


2504 


2521 


2537 


2553 


16 


7 


2651 


2667 


2684 


2700 


2716 


16 


8 


2813 


2830 


2846 


2862 


2878 


16 


9 


2975 


2991 


3008 


3024 


3040 


16 


270 


43136 


43152 


43169 


43185 


43201 


16 


1 


3297 


3313 


3329 


3345 


3361 


16 


2 


3457 


3473 


3489 


3505 


3521 


16 


3 


3616 


3632 


3648 


3664 


3680 


16 


4 


3775 


3791 


3807 


3823 


3838 


16 


5 


43933 


43949 


43965 


43981 


43996 


16 


6 


4091 


4107 


4122 


4138 


4154 


16 


7 


4248 


4264 


4279 


4295 


4311 


16 


8 


4404 


4420 


4436 


4451 


4467 


16 


9, 


4560 


4576 


4592 


4607 


4623 


16 



L 



IW 



OF NUMBERS 



N 


5 


6 


7 


8 


9 


D 


330 


34341 


34361 


34380 


34400 


34420 


30 


1 


4537 


4557 


4577 


4596 


4616 


30 


3 


4733 


4753 


4772 


4792 


4811 


30 


3 


4928 


4947 


4967 


4986 


5005 


19 


4 


5122 


5141 


5160 


5180 


5199 


19 


5 


35315 


35334 


35353 


35372 


35392 


19 


6 


5507 


5526 


5545 


5564 


5583 


19 


7 


5698 


5717 


5736 


5755 


5774 


19 


8 


5889 


5908 


5927 


5946 


5965 


19 


9 


6078 


6097 


6116 


6135 


6154 


19 


330 


36267 


36286 


36305 


36324 


36342 


19 


1 


6455 


6474 


6493 


6511 


6530 


19 


3 


6642 


6661 


6680 


6698 


6717 


19 


3 


6829 


6847 


6866 


6884 


6903 


19 


4 


7014 


7033 


7051 


7070 


7088 


18 


5 


37199 


37218 


37236 


37254 


37273 


18 


6 


7383 


7401 


7420 


7438 


7457 


18 


7 


7566 


7585 


7603 


7621 


7639 


18 


S 


7749 


7767 


7785 


7803 


7822 


18 


9 


7931 


7949 


7967 


7985 


8003 


18 


340 


38112 


38130 


38148 


38166 


38184 


18 


1 


8292 


8310 


8328 


8346 


8364 


18 


3 


8471 


8489 


8507 


8525 


8543 


18 


3 


8650 


8668 


8686 


8703 


8721 


18 


4 


8828 


8846 


8863 


8881 


8899 


18 


5 


39005 


39023 


39041 


39058 


39076 


18 


6 


9182 


9199 


9217 


9235 


9252 


18 


7 


9358 


9375 


9393 


9410 


9428 


18 


8 


9533 


9550 


9568 


9585 


9602 


17 


9 


9707 


9724 


9742 


9759 


9777 


17 


350 


39S81 


39898 


39915 


39933 


39950 


17 


1 


40054 


40071 


40088 


40106 


40123 


17 


3 


0226 


0243 


0261 


0278 


0295 


17 


3 


0398 


0415 


0432 


0449 


0466 


17 


4 


0569 


0586 


0603 


0620 


0637 


17 


5 


40739 


40756 


40773 


40790 


40807 


17 


6 


0909 


0926 


0943 


0960 


0976 


17 


7 


1078 


1095 


1111 


1128 


1145 


17 


8 


1246 


1263 


1280 


1296 


1313 


17 


9 


1414 


1430 


1447 


1464 


1481 


17 


360 


41581 


41597 


41614 


41631 


41647 


17 


1 


1747 


1764 


1780 


1797 


1814 


17 


3 


1913 


1929 


1946 


1963 


1979 


17 


3 


2078 


2095 


2111 


2127 


2144 


16 


4 


2243 


2259 


2275 


2292 


2308 


16 


5 


42406 


42423 


42439 


42455 


42472 


16 


6 


2570 


2586 


2602 


2619 


2635 


16 


7 


2732 


2749 


2765 


2781 


2797 


16 


8 


2894 


2911 


2927 


2943 


2959 


16 


9 


3056 


3072 


3088 


3104 


3120 


16 


370 


43217 


43233 


43249 


43265 


43281 


16 


1 


3377 


3393 


3409 


3425 


3441 


16 


3 


3537 


3553 


3569 


3584 


3600 


16 


3 


3696 


3712 


3727 


3743 


3759 


16 


4 


3854 


3870 


3886 


3902 


3917 


16 


5 


44012 


44028 


44044 


44059 


44075 


16 


6 


4170 


4185 


4201 


4217 


4232 


16 


7 


4326 


4342 


4358 


4373 


4389 


16 


8 


4483 


4498 


4514 


4529 


4545 


16 


9 


4638 


4654 


4669 


4685 


4700 


16 



161 



-^^-s^ 



1. LOGARITHMS 



N 





1 


3 


3 


4 


D 




280 


44716 


44731 


44747 


44762 


44778 


15 




1 


4871 


4886 


4902 


4917 


4932 


15 




2 


5025 


5040 


5056 


5071 


5086 


15 




3 


5179 


5194 


5209 


5225 


5240 


15 




4 


5332 


5347 


5362 


5378 


5393 


15 




5 


45484 


45500 


45515 


45530 


45545 


15 




6 


5637 


5652 


5667 


5682 


5697 


15 




7 


5788 


5803 


5818 


5834 


5849 


15 




8 


5939 


5954 


5969 


5984 


6000 


15 




9 


6090 


6105 


6120 


6135 


6150 


15 




290 


46240 


46255 


46270 


46285 


46300 


15 




1 


6389 


6404 


6419 


6434 


6449 


15 




3 


6538 


6553 


6568 


6583 


6598 


15 




3 


6687 


6702 


6716 


6731 


6746 


15 




4 


6835 


6850 


6864 


6879 


6894 


15 




5 


46982 


46997 


47012 


47026 


47041 


15 




6 


7129 


7144 


7159 


7173 


7188 


15 




7 


7276 


7290 


7305 


7319 


7334 


15 




8 


7422 


7436 


7451 


7465 


7480 


15 




9 


7567 


7582 


7596 


7611 


7625 


14 




300 


47712 


47727 


47741 


47756 


47770 


14 




1 


7857 


7871 


7885 


7900 


7914 


14 




3 


8001 


8015 


8029 


8044 


8058 


14 




3 


8144 


8159 


8173 


8187 


8202 


14 




4 


8287 


8302 


8316 


8330 


8344 


14 




5 


48430 


48444 


48458 


48473 


48487 


14 




6 


8572 


8586 


8601 


8615 


8629 


14 




7 


8714 


8728 


8742 


8756 


8770 


14 




8 


8855 


8869 


8883 


8897 


8911 


14 




9 


8996 


9010 


9024 


9038 


9052 


14 




310 


49136 


49150 


49164 


49178 


49192 


14 




1 


9276 


9290 


9304 


9318 


9332 


14 




3 


9415 


9429 


9443 


9457 


9471 


14 




3 


9554 


9568 


9582 


9596 


9610 


14 




4 


9693 


9707 


9721 


9734 


9748 


14 




5 


49831 


49845 


49859 


49872 


49886 


14 




6 


9969 


9982 


9996 


50010 


50024 


14 




7 


50106 


50120 


50133 


0147 


0161 


14 




8 


0243 


0256 


0270 


0284 


0297 


14 




9 


0379 


0393 


0406 


0420 


0433 


14 




330 


50515 


50529 


50542 


50556 


50569 


14 




1 


0651 


0664 


0678 


0691 


0705 


13 




3 


0786 


0799 


0813 


0826 


0840 


13 




3 


0920 


0934 


0947 


0961 


0974 


13 




4 


1055 


1068 


1081 


1095 


1108 


13 


1 


5 


51188 


51202 


51215 


51228 


51242 


13 


1 


6 


1322 


1335 


1348 


1362 


1375 


13 




7 


1455 


1468 


1481 


1495 


1508 


13 




8 


1587 


1601 


1614 


1627 


1640 


13 




9 


1720 


1733 


1746 


1759 


1772 


13 




330 


51851 


51865 


51878 


51891 


51904 


13 




1 


1983 


1996 


2009 


2022 


2035 


13 




3 


2114 


2127 


2140 


2153 


2166 


13 




3 


2244 


2257 


2270 


2284 


2297 


13 




4 


2375 


2388 


2401 


2414 


2427 


13 




5 


52504 


52517 


52530 


52543 


52556 


13 




6 


2634 


2647 


2660 


2673 


2686 


13 




7 


2763 


2/78 


2789 


2802 


2815 


13 




8 


2892 


2905 


2917 


2930 


2943 


13 




9 


3020 


3033 


3046 


3058 


3071 


13 


' 



162 



OF NUMBERS 



K 


5 


6 


7 


8 


9 


D 


380 


44793 


44809 


44824 


44840 


44855 


15 


1 


4948 


4963 


4979 


4994 


5010 


15 


2 


5102 


5117 


5133 


5148 


5163 


15 


3 


5255 


5271 


5286 


5301 


5317 


15 


4 


5408 


5423 


5439 


5454 


5469 


15 


5 


45561 


45576 


45591 


45606 


45621 


15 


6 


5712 


5728 


5743 


5758 


5773 


15 


7 


5864 


5879 


5894 


5909 


5924 


15 


8 


6015 


6030 


6045 


6060 


6075 


15 


9 


6165 


6180 


6195 


6210 


6225 


15 


390 


46315 


46330 


46345 


46359 


46374 


15 


1 


6464 


6479 


6494 


6509 


6523 


15 


3 


6613 


6627 


6642 


6657 


6672 


15 


3 


6761 


6776 


6790 


6805 


6820 


15 


4 


6909 


6923 


6938 


6953 


6967 


15 


5 


47056 


47070 


47085 


47100 


47114 


15 


6 


7202 


7217 


7232 


7246 


7261 


15 


7 


7349 


7363 


7378 


7392 


7407 


15 


8 


7454 


7509 


7524 


7538 


7553 


15 


9 


7640 


7654 


7669 


7683 


7698 


14 


300 


47784 


47799 


47813 


47828 


47842 


14 


1 


7929 


7943 


7958 


7972 


7986 


14 


3 


8073 


8087 


8101 


8116 


8130 


14 


3 


8216 


8230 


8244 


8259 


8273 


14 


4 


8359 


8373 


8387 


8401 


8416 


14 


5 


48501 


48515 


48530 


48544 


48558 


14 


6 


8643 


8657 


8671 


8686 


8700 


14 


7 


8785 


8799 


8813 


8827 


8841 


14 


8 


8926 


8940 


8954 


8968 


8982 


14 


9 


9066 


9080 


9094 


9108 


9122 


14 


310 


49206 


49220 


49234 


49248 


49262 


14 


1 


9346 


9360 


9374 


9388 


9402 


14 


3 


9485 


9499 


9513 


9527 


9541 


14 


3 


9624 


9638 


9651 


9665 


9679 


14 


4 


9762 


9776 


9790 


9803 


9817 


14 


5 


49900 


49914 


49927 


49941 


49955 


14 


6 


50037 


50051 


50065 


50079 


50092 


14 


7 


0174 


0188 


0202 


0215 


0229 


14 


8 


0311 


0325 


0338 


0352 


0365 


14 


9 


0447 


0461 


0474 


0488 


0501 


14 


330 


50583 


50596 


50610 


50623 


50637 


14 


1 


0718 


0732 


0745 


0759 


0772 


13 


3 


0853 


0866 


0880 


0893 


0907 


13 


3 


0987 


1001 


1014 


1028 


1041 


13 


4 


1121 


1135 


1148 


1162 


1175 


13 


5 


51255 


51268 


51282 


51295 


51308 


13 


6 


1388 


1402 


1415 


1428 


1441 


13 


7 


1521 


1534 


1548 


1561 


1574 


13 


8 


1654 


1667 


1680 


1693 


1706 


13 


9 


1786 


1799 


1812 


1825 


1838 


13 


330 


51917 


51930 


51943 


51957 


51970 


13 


1 


2048 


2061 


2075 


2088 


2101 


13 


3 


2179 


2192 


2205 


2218 


2231 


13 


3 


2310 


2323 


2336 


2349 


2362 


13 


4 


2440 


2453 


2466 


2479 


2492 


13 


5 


52569 


52582 


52595 


52608 


52621 


13 


6 


2699 


2711 


2724 


2737 


2750 


13 


7 


2827 


2840 


2853 


2866 


2879 


13 


8 


2956 


2969 


2982 


2994 


3007 


13 


9 


3084 


3097 


3110 


3122 


3135 13 



163 



I. LOGARITHMS 



N 





1 


3 


3 


4 


D 


340 


53148 


53161 


53173 


53186 


53199 


13 


1 


3275 


3288 


3301 


3314 


3326 


13 


3 


3403 


3415 


3428 


3441 


3453 


13 


3 


3529 


3542 


3555 


3567 


3580 


13 


4 


3656 


3668 


3681 


3694 


3706 


13 


5 


53782 


53794 


53807 


53820 


53832 


13 


6 


3908 


3920 


3933 


3945 


3958 


13 


7 


4033 


4045 


4058 


4070 


4083 


13 


8 


4158 


4170 


4183 


4195 


4208 


13 


9 


4283 


4295 


4307 


4320 


4332 


13 


350 


54407 


54419 


54432 


54444 


54456 


13 


1 


4531 


4543 


4555 


4568 


4580 


13 


2 


4654 


4667 


4679 


4691 


4704 


13 


3 


4777 


4790 


4802 


4814 


4827 


13 


4 


4900 


4913 


4925 


4937 


4949 


13 


6 


55023 


55035 


55047 


55060 


55072 


13 


6 


5145 


5157 


5169 


5182 


5194 


13 


7 


5267 


5279 


5291 


5303 


5315 


13 


8 


5388 


5400 


5413 


5425 


.5437 


13 


9 


5509 


5522 


5534 


5546 


5558 


13 


360 


55630 


55642 


55654 


55666 


65678 


13 


1 


5751 


5763 


5775 


5787 


5799 


13 


3 


5871 


5883 


5895 


5907 


5919 


13 


3 


5991 


6003 


6015 


6027 


6038 


13 


4 


6110 


6122 


6134 


6146 


6158 


13 


5 


56229 


56241 


56253 


56265 


66277 


13 


6 


6348 


6360 


6372 


6384 


6396 


13 




6467 


6478 


6490 


6502 


6514 


13 


8 


6585 


6597 


6608 


6620 


6632 


13 


9 


6703 


6714 


6726 


6738 


6750 


13 


370 


56820 


56832 


56844 


56855 


56867 


13 


1 


6937 


6949 


6961 


6972 


6984 


13 


3 


7054 


7066 


7078 


7089 


7101 


13 


3 


7171 


7183 


7194 


7206 


7217 


13 


4 


7287 


7299 


7310 


7322 


7334 


13 


5 


57403 


67415 


57426 


67438 


57449 


13 


6 


7519 


7530 


7542 


7553 


7565 


13 




7634 


7646 


7657 


7669 


7680 


11 


8 


7749 


7761 


7772 


7784 


7795 


11 


9 


7864 


7875 


7887 


7898 


7910 


11 


380 


57978 


57990 


58001 


58013 


68024 


11 


1 


8092 


8104 


8115 


8127 


8138 


11 


3 


8206 


8218 


8229 


8240 


8252 


11 


3 


8320 


8331 


8343 


8354 


8365 


11 


4 


8433 


8444 


8456 


8467 


8478 


11 


5 


58546 


58557 


58569 


58580 


68591 


11 


6 


8659 


8670 


8681 


8692 


8704 


11 


7 


8771 


8782 


8794 


8805 


8816 


11 


8 


8883 


8894 


8906 


8917 


8928 


11 


9 


8995 


9006 


9017 


9028 


9040 


11 


390 


59106 


59118 


59129 


59140 


59151 


11 


1 


9218 


9229 


9240 


9251 


9262 


11 


3 


9329 


9340 


9351 


9362 


9373 


11 


3 


9439 


9450 


9461 


9472 


9483 


11 


4 


9550 


9561 


9572 


9583 


9594 


11 


5 


59660 


59671 


59682 


59693 


69704 


11 


6 


9770 


9780 


9791 


9802 


9813 


11 


7 


9879 


9890 


9901 


9912 


9923 


11 


8 


9988 


9999 


60010 


60021 


60032 


11 


9 


60097 


60108 


60119 


60130 


60141 


11 



164 



OP NUMBERS 



N 


5 


6 


7 


8 


9 


D 


340 


53212 


53224 


53237 


53250 


53263 


13 


1 


3339 


3352 


3364 


3377 


3390 


13 


3 


3466 


3479 


3491 


3504 


3517 


13 


3 


3593 


3605 


3618 


3631 


3643 


13 


4 


3719 


3732 


3744 


3757 


3769 


13 


5 


53845 


53857 


53870 


53882 


53895 


13 


6 


3970 


3983 


3995 


4008 


4020 


13 


7 


4095 


4108 


4120 


4133 


4145 


13 


8 


4220 


4233 


4245 


4258 


4270 


13 


9 


4345 


4357 


4370 


4382 


4394 


13 


350 


54469 


54481 


54494 


54506 


54518 


13 


1 


4593 


4605 


4617 


4630 


4642 


13 


2 


4716 


4728 


4741 


4753 


4765 


13 


3 


4839 


4851 


4864 


4876 


4888 


13 


4 


4962 


4974 


4986 


4998 


5011 


13 


5 


55084 


55096 


55108 


55121 


55133 


13 


6 


5206 


5218 


5230 


5242 


5255 


13 


7 


5328 


5340 


5352 


5364 


5376 


13 


8 


5449 


5461 


5473 


5485 


5497 


13 


9 


5570 


5582 


5594 


5606 


5618 


13 


360 


55691 


55703 


55715 


55727 


55739 


13 


1 


5811 


5823 


5835 


5847 


5859 


13 


2 


5931 


5943 


5955 


5967 


5979 


13 


3 


6050 


6062 


6074 


6086 


6098 


13 


4 


6170 


6182 


6194 


6205 


6217 


13 


5 


56289 


56301 


56312 


56324 


56336 


13 


6 


6407 


6419 


6431 


6443 


6455 


13 


7 


6526 


6538 


6549 


6561 


6573 


13 


8 


6644 


6656 


6667 


6679 


6691 


13 


9 


6761 


6773 


6785 


6797 


6808 


13 


370 


56879 


56891 


56902 


56914 


56926 


13 


1 


6996 


7008 


7019 


7031 


7043 


13 


2 


7113 


7124 


7136 


7148 


7159 


13 


3 


7229 


7241 


7252 


7264 


7276 


13 


4 


7345 


7357 


7368 


7380 


7392 


13 


5 


57461 


57473 


57484 


57496 


57507 


13 


6 


7576 


7588 


7600 


7611 


7623 


13 


7 


7692 


7703 


7715 


7726 


7738 


11 


8 


7807 


7818 


7830 


7841 


7852 


11 


9 


7921 


7933 


7944 


7955 


7967 


11 


380 


58035 


58047 


5S058 


58070 


58081 


11 


1 


8149 


8161 


8172 


8184 


8195 


11 


2 


8263 


8274 


8286 


8297 


8309 


11 


3 


8377 


8388 


8399 


8410 


8422 


11 


4 


8490 


8501 


8512 


8524 


8535 


11 


5 


58602 


58614 


58625 


58636 


58647 


11 


6 


8715 


8726 


8737 


8749 


8760 


11 


7 


8827 


8838 


8850 


8861 


8872 


11 


8 


8939 


8950 


8961 


8973 


8984 


11 


9 


9051 


9062 


9073 


9084 


9095 


11 


390 


59162 


59.173 


59184 


59195 


59207 


11 


1 


9273 


9284 


9295 


9306 


9318 


11 


2 


9384 


9395 


9406 


9417 


9428 


11 


3 


9494 


9506 


9517 


9528 


9539 


11 


4 


9605 


9616 


9627 


9638 


9649 


11 


5 


59715 


59726 


59737 


59748 


59759 


11 


6 


9824 


9835 


9846 


9857 


9868 


11 


7 


9934 


9945 


9956 


9966 


9977 


11 


8 


60043 


60054 


60065 


60076 


60086 


11 


9 


60152 


60163 


60173 


60184 


60195 


11 



165 



1. LOGARITHMS 



N 





1 


2 


3 


4 


D 


400 


60206 


60217 


60228 


60239 


60249 


11 


1 


0314 


0325 


0336 


0347 


0358 


11 


3 


0423 


0433 


0444 


0455 


0466 


11 


3 


0531 


0541 


0552 


0563 


0574 


11 


4 


0638 


0649 


0660 


0670 


0681 


11 


5 


60746 


60756 


60767 


60778 


60788 


11 


6 


0853 


0863 


0874 


0885 


0895 


11 


7 


0959 


0970 


0981 


0991 


1002 


11 


8 


1066 


1077 


1087 


1098 


1109 


11 


9 


1172 


1183 


1194 


1204 


1215 


11 


410 


61278 


61289 


61300 


61310 


61321 


11 


1 


1384 


1395 


1405 


1416 


1426 


11 


2 


1490 


1500 


1511 


1521 


1532 


11 


3 


1595 


1606 


1616 


1627 


1637 


10 


4 


1700 


1711 


1721 


1731 


1742 


10 


5 


61805 


61815 


61826 


61836 


61847 


10 


6 


1909 


1920 


1930 


1941 


1951 


10 


7 


2014 


2024 


2034 


2045 


2055 


10 


8 


2118 


2128 


2138 


2149 


2159 


10 


9 


2221 


2232 


2242 


2252 


2263 


10 


430 


62325 


62335 


62346 


62356 


62366 


10 


1 


2428 


2439 


2449 


2459 


2469 


10 


2 


2531 


2542 


2552 


2562 


2572 


10 


3 


2634 


2644 


2655 


2665 


2675 


10 


4 


2737 


2747 


2757 


2767 


2778 


10 


5 


62839 


62849 


62859 


62870 


62880 


10 


6 


2941 


2951 


2961 


2972 


2982 


10 


7 


3043 


3053 


3063 


3073 


3083 


10 


S 


3144 


3155 


3165 


3175 


3185 


10 


9 


3246 


3256 


3266 


3276 


3286 


10 


430 


63347 


63357 


63367 


63377 


63387 


10 


1 


3448 


3458 


3468 


3478 


3488 


10 


2 


3548 


3558 


3568 


3579 


3589 


10 


3 


3649 


3659 


3669 


3679 


3689 


10 


4 


3749 


3759 


3769 


3779 


3789 


10 


5 


63849 


63859 


63869 


63879 


63889 


10 


6 


3949 


3959 


3969 


3979 


3988 


10 


7 


4048 


4058 


4068 


4078 


4088 


10 


8 


4147 


4157 


4167 


4177 


4187 


10 


9 


4246 


4256 


4266 


4276 


4286 


10 


440 


64345 


64355 


64365 


64375 


64385 


10 


1 


4444 


4454 


4464 


4473 


4483 


10 


2 


4542 


4552 


4562 


4572 


4582 


10 


3 


4640 


4650 


4660 


4670 


4680 


10 


4 


4738 


4748 


4758 


4768 


4777 


10 


5 


64836 


64846 


64856 


64865 


64875 


10 


6 


4933 


4943 


4953 


4963 


4972 


10 


7 


5031 


5040 


5050 


5060 


5070 


10 


8 


5128 


5137 


5147 


5157 


5167 


10 


9 


5225 


5234 


5244 


5254 


5263 


10 


450 


65321 


65331 


65341 


65350 


65360 


10 


1 


5418 


5427 


5437 


5447 


5456 


10 


2 


5514 


5523 


5533 


5543 


6552 


10 


3 


5610 


5619 


6629 


6639 


5648 


10 


4 


5706 


5715 


5725 


6734 


5744 


10 


5 


65801 


65811 


65820 


65830 


65839 


10 


6 


5896 


5906 


5916 


5925 


5935 


10 


7 


5992 


6001 


6011 


6020 


6030 


9 


8 


6087 


6096 


6106 


6115 


6124 


9 


9 


6181 


6191 


6200 


6210 


6219 


9 



L 



166 



OF NUMBERS 



N 


5 


6 


7 


8 


9 


D 


400 


60260 


60271 


60282 


60293 


60304 


11 


1 


0369 


0379 


0390 


0401 


0412 


11 


2 


0477 


0487 


0498 


0509 


0520 


11 


3 


0584 


0595 


0606 


0617 


0627 


11 


4 


0692 


0703 


0713 


0724 


0735 


11 


5 


60799 


60810 


60821 


60831 


60842 


11 


6 


0906 


0917 


0927 


0938 


0949 


11 


7 


1013 


1023 


1034 


1045 


1055 


11 


8 


1119 


1130 


1140 


1151 


1162 


11 


9 


1225 


1236 


1247 


1257 


1268 


11 


410 


61331 


61342 


61352 


61363 


61374 


11 


1 


1437 


1448 


1458 


1469 


1479 


11 


2 


1542 


1553 


1563 


1574 


1584 


11 


3 


1648 


1658 


1669 


1679 


1690 


10 


4 


1752 


1763 


1773 


1784 


1794 


10 


5 


61857 


61868 


61878 


61888 


61899 


10 


6 


1962 


1972 


1982 


1993 


2003 


10 


7 


2066 


2076 


2086 


2097 


2107 


10 


8 


2170 


2180 


2190 


2201 


2211 


10 


9 


2273 


2284 


2294 


2304 


2315 


10 


420 


62377 


62387 


62397 


62408 


62418 


10 


1 


2480 


2490 


2500 


2511 


2521 


10 


2 


2583 


2593 


2603 


2613 


2624 


10 


3 


2685 


2696 


2706 


2716 


2726 


10 


4 


2788 


2798 


2808 


2818 


2829 


10 


/> 


62890 


62900 


62910 


62921 


62931 


10 


6 


2992 


3002 


3012 


3022 


3033 


10 


7 


3094 


3104 


3114 


3124 


3134 


10 


8 


3195 


3205 


3215 


3225 


3236 


10 


9 


3296 


3306 


3317 


3327 


3337 


10 


430 


63397 


63407 


63417 


63428 


63438 


10 


1 


3498 


3508 


3518 


3528 


3538 


10 


2 


3599 


3609 


3619 


3629 


3639 


10 


3 


3699 


3709 


3719 


3729 


3739 


10 


4 


3799 


3809 


3819 


3829 


3839 


10 


5 


63899 


63909 


63919 


63929 


63939 


10 


6 


3998 


4008 


4018 


4028 


4038 


10 


7 


4098 


4108 


4118 


4128 


4137 


10 


8 


4197 


4207 


4217 


4227 


4237 


10 


9 


4296 


4306 


4316 


4326 


4335 


10 


440 


64395 


64404 


64414 


64424 


64434 


10 


1 


4493 


4503 


4513 


4523 


4532 


10 


2 


4591 


4601 


4611 


4621 


4631 


10 


3 


4689 


4699 


4709 


4719 


4729 


10 


4 


4787 


4797 


4807 


4816 


4826 


10 


5 


64885 


64895 


64904 


64914 


64924 


10 


6 


4982 


4992 


5002 


5011 


5021 


10 


7 


5079 


5089 


5099 


5108 


5118 


10 


8 


5176 


5186 


5196 


5205 


5215 


10 


9 


5273 


5283 


5292 


5302 


5312 


10 


450 


65369 


65379 


65389 


65398 


65408 


10 


1 


5466 


5475 


5485 


5495 


5504 


10 


2 


5562 


5571 


5581 


5591 


5600 


10 


3 


5658 


5667 


5677 


5686 


5696 


10 


4 


5753 


5763 


5772 


5782 


5792 


10 


5 


65849 


65858 


65868 


65877 


65887 


10 


6 


5944 


5954 


5963 


5973 


5982 


10 i 


7 


6039 


6049 


6058 


6068 


6077 


9 


8 


6134 


6143 


6153 


6162 


6172 


d 1 


9 


6229 


6238 


6247 


6257 


6266 


9 1 



167 



I. LOGARITHMS 



N 





1 


2 


3 


4 


D 


460 


66276 


66285 


66295 


06304 


66314 


9 


1 


6370 


6380 


6389 


6398 


6408 


9 


2 


6464 


6474 


6483 


6492 


6502 


9 


3 


6558 


6567 


6577 


6586 


6596 


9 


4 


6652 


6661 


6671 


6680 


6689 


9 


5 


66745 


66755 


66764 


60773 


66783 


9 


6 


6839 


6848 


6857 


6867 


6876 


9 


7 


6932 


6941 


6950 


6960 


6969 


9 


8 


7025 


7034 


7043 


7052 


7062 


9 


9 


7117 


7127 


7136 


7145 


7154 


9 


470 


67210 


67219 


67228 


67237 


67247 


9 


1 


7302 


7311 


7321 


7330 


7339 


9 


2 


7394 


7403 


7413 


7422 


7431 


9 


3 


7486 


7495 


7504 


7514 


7523 


9 


4 


7578 


7587 


7596 


7605 


7614 


9 


5 


67669 


67679 


67688 


67697 


67706 


9 
9 


6 


7761 


7770 


7779 


7788 


7797 


7 


7852 


7861 


7870 


7879 


7888 


9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 


8 


7943 


7952 


7961 


7970 


7979 


9 


8034 


8043 


8052 


8061 


8070 


480 


68124 


68133 


68142 


68151 


68160 


1 


8215 


8224 


8233 


8242 


8251 


2 


8305 


8314 


8323 


8332 


8341 


3 


8395 


8404 


8413 


8422 


8431 


4 


8485 


8494 


8502 


8511 


8520 


5 


68574 


68583 


68592 


68601 


68610 


6 


8664 


8673 


8681 


8690 


8699 




8753 


8762 


8771 


8780 


8789 


8 


8842 


8851 


8860 


8869 


8878 


9 


8931 


8940 


8949 


8958 


8966 


490 


69020 


69028 


69037 


69046 


69055 


1 


9108 


9117 


9126 


9135 


9144 


3 


9197 


9205 


9214 


9223 


9232 


3 


9285 


9294 


9302 


9311 


9320 


4 


9373 


9381 


9390 


9399 


9408 


5 


69461 


69469 


69478 


69487 


69496 


9 


6 


9548 


9557 


9566 


9574 


9583 


9 


7 


9636 


9644 


9653 


9662 


9671 


9 


8 


9723 


9732 


9740 


9749 


9758 


9 


9 


9810 


9819 


9827 


9836 


9845 


9 


600 


69897 


69906 


69914 


69923 


69932 


9 


1 


9984 


9992 


70001 


70010 


70018 


9 


2 


70070 


70079 


0088 


0096 


0105 


9 


3 


0157 


0165 


0174 


0183 


0191 


9 


4 


0243 


0252 


0260 


0269 


0278 


9 


5 


70329 


70338 


70346 


70355 


70364 


9 


6 


0415 


0424 


0432 


0441 


0449 


9 


7 


0501 


0509 


0518 


0526 


0535 


9 


8 


0586 


0595 


0603 


0612 


0621 


9 


9 


0672 


0680 


0689 


0697 


0706 


9 


610 


70757 


70766 


70774 


70783 


70791 


8 


1 


0842 


0851 


0859 


0868 


0876 


8 


2 


0927 


0935 


0944 


0952 


0961 


8 


3 


1012 


1020 


1029 


1037 


1046 


8 


4 


1096 


1105 


1113 


1122 


1130 


8 


6 


71181 


71189 


71198 


71206 


71214 


8 


6 


1265 


1273 


1282 


1290 


1299 


8 


7 


1349 


1357 


1360 


1374 


1383 


8 


8 


1433 


1441 


1450 


1458 


1460 


8 


9 


1517 


1525 


1533 


1542 


1550 


8 



168 



OF NUMBERS 



N 


5 


6 


7 


8 


9 


D 


460 


66323 


66332 


66342 


66351 


66361 


~9 


1 


6417 


6427 


6436 


6445 


6455 


9 


2 


6511 


6521 


6530 


6539 


6549 


9 


3 


6605 


6614 


6624 


6633 


6642 


9 


4 


6699 


6708 


6717 


6727 


6736 


9 


5 


66792 


66801 


66811 


66820 


66829 


9 


6 


6885 


6894 


6904 


6913 


6922 


9 


7 


6978 


6987 


6997 


7006 


7015 


9 


8 


7071 


7080 


7089 


7099 


7108 


9 


9 


7164 


7173 


7182 


7191 


7201 


9 


470 


67256 


67265 


67274 


67284 


67293 


9 


1 


7348 


7357 


7367 


7376 


7385 


9 


2 


7440 


7449 


7459 


7468 


7477 


9 


3 


7532 


7541 


7550 


7560 


7569 


9 


4 


7624 


7633 


7642 


7651 


7660 


9 


5 


67715 


67724 


67733 


67742 


67752 


9 


6 


■ 7806 


7815 


7825 


7834 


7843 


9 


7 


7897 


7906 


7916 


7925 


7934 


9 


8 


7988 


7997 


8006 


8015 


8024 


9 


9 


8079 


8088 


8097 


8106 


8115 


9 


480 


68169 


68178 


68187 


68196 


68205 


9 


1 


8260 


8269 


8278 


8287 


8296 


9 


2 


8350 


8359 


8368 


8377 


8386 


9 


3 


8440 


8449 


8458 


8467 


8476 


9 


4 


8529 


8538 


8547 


8556 


8565 


9 


5 


68619 


68628 


68637 


68646 


68655 


9 


6 


8708 


8717 


8726 


8735 


8744 


9 


7 


8797 


8806 


8815 


8824 


8833 


9 


8 


8886 


8895 


8904 


8913 


8922 


9 


9 


8975 


8984 


8993 


9002 


9011 


9 


490 


69064 


69073 


69082 


69090 


69099 


9 


1 


9152 


9161 


9170 


9179 


9188 


9 


2 


9241 


9249 


9258 


9267 


9276 


9 


3 


9329 


9338 


9346 


9355 


9364 


9 


4 


9417 


9425 


9434 


9443 


9452 


9 


5 


69504 


69513 


69522 


69531 


69539 


9 


6 


9592 


9601 


9609 


9618 


9627 


9 


7 


9679 


9688 


9697 


9705 


9714 


9 


8 


9767 


9775 


9784 


9793 


9801 


9 


9 


9854 


9862 


9871 


9880 


9888 


9 


500 


69940 


69949 


69958 


69966 


69975 


9 


1 


70027 


70036 


70044 


70053 


70062 


9 


2 


0114 


0122 


0131 


0140 


0148 


9 


3 


0200 


0209 


0217 


0226 


0234 


9 


4 


0286 


0295 


0303 


0312 


0321 


9 


5 


70372 


70381 


70389 


70398 


70406 


9 


6 


0458 


0467 


0475 


0484 


0492 


9 


7 


0544 


0552 


0561 


0569 


0578 


9 


8 


0629 


0638 


0646 


0655 


0663 


9 


9 


0714 


0723 


0731 


0740 


0749 




510 


70800 


70808 


70817 


70825 


70834 


8 


1 


0885 


0893 


0902 


0910 


0919 


g 


2 


0969 


0978 


0986 


0995 


1003 


8 


3 


1054 


1003 


1071 


1079 


1088 


8 


4 


1139 


1147 


1155 


1164 


1172 


8 


5 


71223 


71231 


71240 


71248 


71257 


8 


6 


1307 


1315 


1324 


1332 


1341 


8 


7 


1391 


1399 


1408 


1416 


1425 


8 


8 


1475 


1483 


1492 


1500 


1508 


8 


9 


1559 


1567 


1575 


1584 


1592 


8 



xm 



I. LOGARITHMS 



N 





1 


2 


3 


4 


D 

8 


620 


71600 


71609 


71617 


71625 


71634 


1 


1684 


1692 


1700 


1709 


1717 


8 


2 


1767 


1775 


1784 


1792 


1800 


8 


3 


1850 


1858 


1867 


1875 


1883 


8 


4 


1933 


1941 


1950 


1958 


1966 


8 


5 


72016 


72024 


72032 


72041 


72049 


8 


6 


2099 


2107 


2115 


2123 


2132 


8 


7 


2181 


2189 


2198 


2206 


2214 


8 


8 


2263 


2272 


2280 


2288 


2296 


8 


9 


2346 


2354 


2362 


2370 


2378 


8 


630 


72428 


72436 


72444 


72452 


72460 


8 


1 


2509 


2518 


2526 


2534 


2542 


8 


2 


2591 


2599 


2607 


2616 


2624 


8 


3 


2673 


2681 


2689 


2697 


2705 


8 


4 


2754 


2762 


2770 


2779 


,2787 


8 


6 


72835 


72843 


72852 


72860 


72868 


8 


6 


2916 


2925 


2933 


2941 


2949 


8 


7 


2997 


3006 


3014 


3022 


3030 


8 


8 


3078 


3086 


3094 


3102 


3111 


8 


9 


3159 


3167 


3175 


3183 


3191 


8 


640 


73239 


73247 


73255 


73263 


73272 


8 


1 


3320 


3328 


3336 


3344 


3352 


8 


3 


3400 


3408 


3416 


3424 


3432 


8 


3 


3480 


3488 


3496 


3504 


3512 


8 


4 


3560 


3568 


3576 


3584 


3592 


8 


6 


73640 


73648 


73656 


73664 


73672 


8 


6 


3719 


3727 


3735 


3743 


3751 


8 


7 


3799 


3807 


3815 


3823 


3830 


8 


8 


3878 


3886 


3894 


3902 


3910 


8 


9 


3957 


3965 


3973 


3981 


3989 


8 


650 


74036 


74044 


74052 


74060 


74068 


8 


1 


4115 


4123 


4131 


4139 


4147 


8 


2 


4194 


4202 


4210 


4218 


4225 


8 


3 


4273 


4280 


4288 


4296 


4304 


8 


4 


4351 


4359 


4367 


4374 


4382 


8 


6 


74429 


74437 


74445 


74453 


74461 


8 


6 


4507 


4515 


4523 


4531 


4539 


8 


7 


4586 


4593 


4601 


4609 


4617 


8 


8 


4663 


4671 


4679 


4687 


4695 


8 


9 


4741 


4749 


4757 


4764 


4772 


8 


660 


74S19 


74827 


74834 


74842 


74850 


8 


1 


4896 


4904 


4912 


4920 


4927 


8 


2 


4974 


4981 


4989 


4997 


5005 


8 


3 


5051 


5059 


5066 


5074 


5082 


8 


4 


5128 


6136 


5143 


5151 


6159 


8 


6 


75205 


75213 


75220 


75228 


75236 


8 


6 


5282 


6289 


6297 


6305 


6312 


8 


7 


6358 


6366 


6374 


5381 


6389 


8 


8 


5435 


5442 


6450 


6458 


5465 


8 


9 


5511 


6519 


6526 


5534 


6542 


8 


570 


75587 


75595 


75603 


75610 


75618 


8 


1 


6664 


6671 


6679 


6686 


6694 


8 


2 


6740 


6747 


6755 


6762 


6770 


8 


3 


6815 


6823 


6831 


6838 


6846 


8 


4 


6891 


6899 


6906 


5914 


5921 


8 


6 


75967 


75974 


759cS2 


75989 


75997 


8 


6 


6042 


6050 


6057 


6065 


6072 


8 


7 


6118 


6125 


6133 


6140 


6148 


8 


8 


6193 


6200 


6208 


6215 


6223 


7 


9 


6268 


6275 


6283 


6290 


6298 


7 



L 



170 



OF NUMBERS 



N 


5 


6 


7 


8 


9 


D 


520 


71642 


71650 


71659 


71667 


71675 


8 


1 


1725 


1734 


1742 


1750 


1759 


8 


3 


1809 


1817 


1825 


1834 


1842 


8 


3 


1892 


1900 


1908 


1917 


1925 


8 


4 


1975 


1983 


1991 


1999 


2008 


8 


5 


72057 


72066 


72074 


72082 


72090 




6 


2140 


2148 


2156 


2165 


2173 


g 


7 


2222 


2230 


2239 


2247 


2255 


8 


8 


2304 


2313 


2321 


2329 


2337 


8 


9 


2387 


2395 


2403 


2411 


2419 


8 


530 


72469 


72477 


72435 


72493 


72501 




1 


2550 


2558 


2567 


2575 


2583 


jl 


2 


2632 


2640 


2648 


2656 


2665 


8 


3 


2713 


2722 


2730 


2738 


2746 


8 


4 


2795 


2803 


2811 


2819 


2827 


8 


5 


72876 


72884 


72892 


72900 


72908 




6 


2957 


2965 


2973 


2981 


2989 


g 


7 




3046 


3054 


3062 


3070 


8 


8 


3119 


3127 


3135 


3143 


3151 


8 


9 


3199 


3207 


3215 


3223 


3231 


8 


540 


73280 


73288 


73296 


73304 


73312 




1 


3360 


3368 


3376 


3384 


3392 


g 


2 


3440 


3448 


3456 


3464 


3472 


g 


3 


3520 


3528 


3536 


3544 


3552 


g 


4 


3600 


3608 


3616 


3624 


3632 


S 


5 


73679 


73687 


73695 


73703 


73711 


d 


6 


3759 


3767 


3775 


3783 


3791 


8 


7 


3838 


3846 


3854 


3862 


3870 


8 


8 


3918 


3926 


3933 


3941 


3949 


8 


9 


3997 


4005 


4013 


4020 


4028 


8 


550 


74076 


74084 


74092 


74099 


74107 


8 


1 


4155 


4162 


4170 


4178 


4186 




2 


4233 


4241 


4249 


4257 


4265 


g 


3 


4312 


4320 


4327 


4335 


4343 


8 


4 


4390 


4398 


4406 


4414 


4421 


8 


5 


7446S 


74476 


74484 


74492 


74500 


8 


6 


4547 


4554 


4562 


4570 


4578 


8 


7 


4624 


4632 


4640 


4648 


4656 




8 


4702 


4710 


4718 


4726 


4733 


8 


9 


4780 


4788 


4796 


4803 


4811 


8 


560 


74858 


74865 


74873 


74881 


74889 


8 


1 


4935 


4943 


4950 


4958 


4966 


8 


2 


5012 


5020 


5028 


5035 


5043 




3 


5089 


5097 


5105 


5113 


5120 


g 


4 


5166 


5174 


5182 


5189 


5197 


8 


5 


75243 


75251 


75259 


75266 


75274 


8 


6 


5320 


5328 


5335 


5343 


5351 


8 


7 


5397 


5404 


5412 


5420 


5427 


8 


8 


5473 


5481 


5488 


5496 


5504 


8 


9 


5549 


5557 


5565 


5572 


5580 


8 


570 


75626 


75633 


75641 


75648 


75656 


8 


1 


5702 


5709 


5717 


5724 


5732 


8 


2 


5778 


5785 


6793 


5800 


5808 


8 


3 


5853 


5861 


5868 


5876 


6884 




4 


5929 


5937 


5944 


5952 


5959 


g 


5 


76005 


76012 


76020 


76027 


76035 


8 


6 


6080 


6087 


6095 


6103 


6110 


8 


7 


6155 


6163 


6170 


6178 


6185 


8 


8 


6230 


6238 


6245 


6253 


6260 


7 


9 


6305 


6313 


6320 


6328 


6335 


7 



17X 



I. LOGARITHMS 



N 





1 


2 


3 


4 


D 


680 


76343 


76350 


76358 


76365 


76373 


~7 


1 


6418 


6425 


6433 


6440 


6448 


7 


2 


6492 


6500 


6507 


6515 


6522 


7 


3 


6567 


6574 


6582 


6589 


6597 


7 


4 


6641 


6649 


6656 


6664 


6671 


7 


5 


76716 


76723 


76730 


76738 


76745 


7 


6 


6790 


6797 


6805 


6812 


6819 


7 


7 


6864 


6871 


6879 


6886 


6893 


7 


8 


6938 


6945 


6953 


6960 


6967 


7 


9 


7012 


7019 


7026 


7034 


7041 


7 


690 


77085 


77093 


77100 


77107 


77115 


7 


1 


7159 


7166 


7173 


7181 


7188 


7 


2 


7232 


7240 


7247 


7254 


7262 


7 


3 


7305 


7313 


7320 


7327 


7335 


7 


4 


7379 


7386 


7393 


7401 


7408 


7 


6 


77452 


77459 


77466 


77474 


77481 


7 


6 


7525 


7532 


7539 


7546 


7554 


7 


7 


7597 


7605 


7612 


7619 


7627 


7 


8 


7670 


7677 


7685 


7692 


7699 


7 


9 


7743 


7750 


7757 


7764 


7772 


7 


600 


77815 


77822 


77830 


77837 


77844 


7 


1 


7887 


7895 


7902 


7909 


7916 


7 


2 


7960 


7967 


7974 


7981 


7988 


7 


3 


8032 


8039 


8046 


8053 


8061 


7 


4 


8104 


8111 


8118 


8125 


8132 


7 


6 


78176 


78183 


78190 


78197 


78204 


7 


6 


8247 


8254 


8262 


8269 


8276 


7 


7 


8319 


8326 


8333 


8340 


8347 


7 


8 


8390 


8398 


8405 


8412 


8419 


7 


9 


8462 


8469 


8476 


8483 


8490 


7 


610 


78533 


78540 


78547 


78554 


78561 


7 


1 


8604 


8611 


8618 


8625 


8633 


7 


2 


8675 


8682 


8689 


8696 


8704 


7 


3 


8746 


8753 


8760 


8767 


8774 


7 


4 


8817 


8824 


8831 


8838 


8845 


7 


6 


78888 


78895 


78902 


78909 


78916 


7 


6 


8958 


8965 


8972 


8979 


8986 


7 


7 


9029 


9036 


9043 


9050 


9057 


7 


8 


9099 


9106 


9113 


9120 


9127 


7 


9 


9169 


9176 


9183 


9190 


9197 


7 


620 


79239 


79246 


79253 


79260 


79267 


7 


1 


9309 


9316 


9323 


9330 


9337 


7 


2 


9379 


9386 


9393 


9400 


9407 


7 


3 


9449 


9456 


9463 


9470 


9477 


7 


4 


9518 


9525 


9532 


9539 


9546 


7 


6 


79588 


79595 


79602 


79609 


79616 


7 


6 


9657 


9664 


9671 


9678 


9685 


7 


7 


9727 


9734 


9741 


9748 


9754 


7 


8 


9796 


9803 


9810 


9817 


9824 


7 


9 


9865 


9872 


9879 


9886 


9893 


7 


630 


79934 


79941 


79948 


79955 


79962 


7 


1 


80003 


80010 


80017 


80024 


80030 


7 


2 


0072 


0079 


0085 


0092 


0099 


7 


3 


0140 


0147 


0154 


0161 


0168 


7 


4 


0209 


0216 


0223 


0229 


0236 


7 


6 


80277 


80284 


80291 


80298 


80305 


7 


6 


0346 


0353 


0359 


0366 


0373 


7 


7 


0414 


0421 


0428 


0434 


0441 


7 


8 


0482 


0489 


0496 


0502 


0509 


7 


9 


0550 


0557 


0564 


0570 


0577 


7 



m 



OF NUMBERS 



N 


5 


6 


7 


8 


9 


D 


580 


76380 


76388 


76395 


76403 


76410 


7 


1 


6455 


6462 


6470 


6477 


6485 


7 


2 


6530 


6537 


6545 


6552 


6559 


7 


3 


6604 


6612 


6619 


6626 


6634 


7 


4 


6678 


6686 


6693 


6701 


6708 


7 


5 


76753 


76760 


76768 


76775 


76782 


7 


6 


6827 


6834 


6842 


6849 


6856 


7 


7 


6901 


6908 


6916 


6923 


6930 


7 


8 


6975 


6982 


6989 


6997 


7004 


7 


9 


7048 


7056 


7063 


7070 


7078 


7 


690 


77122 


77129 


77137 


77144 


77151 


7 


1 


7195 


7203 


7210 


7217 


7225 


7 


2 


7269 


7276 


7283 


7291 


7298 


7 


3 


7342 


7349 


7357 


7364 


7371 


7 


4 


7415 


7422 


7430 


7437 


7444 


7 


5 


77488 


77495 


77503 


77510 


77517 


7 


6 


7561 


7568 


7576 


7583 


7590 


7 


7 


7634 


7641 


7648 


7656 


7663 


7 


8 


7706 


7714 


7721 


7728 


7735 


7 


9 


7779 


7786 


7793 


7801 


7808 


7 


600 


77851 


77859 


77866 


77873 


77880 


7 


1 


7924 


7931 


7938 


7945 


7952 


7 


3 


7996 


8003 


8010 


8017 


8025 


7 


3 


8068 


8075 


8082 


8089 


8097 


7 


4 


8140 


8147 


8154 


8161 


8168 


7 


5 


78211 


78219 


78226 


78233 


78240 


7 


6 


8283 


8290 


8297 


8305 


8312 


7 


7 


8355 


8362 


8369 


8376 


8383 


7 


8 


8426 


8433 


8440 


8447 


8455 


7 


9 


8497 


8504 


8512 


8519 


8526 


7 


610 


78569 


78576 


78583 


78590 


78597 


7 


1 


8640 


8647 


8654 


8661 


8668 


7 


3 


8711 


8718 


8725 


8732 


8739 


7 


3 


8781 


8789 


8796 


8803 


8810 


7 


4 


8852 


8859 


8866 


8873 


8880 


7 


5 


78923 


78930 


78937 


78944 


78951 


7 


6 


8993 


9000 


9007 


9014 


9021 


7 


7 


9064 


9071 


9078 


9085 


9092 


7 


8 


9134 


9141 


9148 


9155 


9162 


7 


9 


9204 


9211 


9218 


9225 


9232 


7 


620 


79274 


79281 


79288 


79295 


79302 


7 


1 


9344 


9351 


9358 


9365 


9372 


7 


3 


9414 


9421 


9428 


9435 


9442 


7 


3 


9484 


9491 


9498 


9505 


9511 


7 


4 


9553 


9560 


9567 


9574 


9581 


7 


5 


79623 


79630 


79637 


79644 


79650 


7 


6 


9692 


9699 


9706 


9713 


9720 


7 


7 


9761 


9768 


9775 


9782 


9789 


7 


8 


9831 


9837 


9844 


9851 


9858 


7 


9 


9900 


9906 


9913 


9920 


9927 


7 


630 


79969 


79975 


79982 


79989 


79996 


7 


1 


80037 


80044 


80051 


80058 


80065 


7 


3 


0106 


0113 


0120 


0127 


0134 


7 


3 


0175 


0182 


0188 


0195 


0202 


7 


4 


0243 


0250 


0257 


0264 


0271 


7 


5 


80312 


80318 


80325 


80332 


80339 


7 


6 


0380 


0387 


0393 


0400 


0407 


7 


7 


0448 


0455 


0462 


0468 


0475 


7 


8 


0516 


0523 


0530 


0536 


0543 


7 


9 


0584 


0591 


0598 


0604 


0611 


7 



173 



J 



1. LOGARITHMS 



N 





1 


3 


3 


4 


D 


640 


80618 


80625 


80632 


80638 


80645 


7 


1 


0686 


0693 


0699 


0706 


0713 


7 


2 


0754 


0760 


0767 


0774 


0781 


7 


3 


0821 


0828 


0835 


0841 


0848 


7 


4 


0889 


0895 


0902 


0909 


0916 


7 


5 


80956 


80963 


80969 


80976 


80983 


7 


6 


1023 


1030 


1037 


1043 


1050 


7 


7 


1090 


1097 


1104 


nil 


1117 


7 


8 


1158 


1164 


1171 


1178 


1184 


7 


9 


1224 


1231 


1238 


1245 


1251 


7 


650 


81291 


81298 


81305 


81311 


81318 


7 


1 


1358 


1365 


1371 


1378 


1385 


7 


2 


1425 


1431 


1438 


1445 


1451 


7 


3 


1491 


1498 


1505 


1511 


1518 


7 


4 


1558 


1564 


1571 


1578 


1584 


7 


6 


81624 


81631 


81637 


81644 


81651 


7 


6 


1690 


1697 


1704 


1710 


1717 


7 




1757 


1763 


1770 


1776 


1783 


7 


s 


1823 


1829 


1836 


1842 


1849 


7 


9 


1889 


1895 


1902 


1908 


1915 


7 


660 


81954 


81961 


81968 


81974 


81981 


7 


1 


2020 


2027 


2033 


2040 


2046 


7 


2 


2086 


2092 


2099 


2105 


2112 


7 


3 


2151 


2158 


2164 


2171 


2178 


7 


4 


2217 


2223 


2230 


2236 


2243 


7 


5 


82282 


82289 


82295 


82302 


82308 


7 


6 


2347 


2354 


2360 


2367 


2373 


7 


7 


2413 


2419 


2426 


2432 


2439 


6 


8 


2478 


2484 


2491 


2497 


2504 


6 


9 


2543 


2549 


2556 


2562 


2569 


6 


670 


82607 


82614 


82620 


82627 


82633 


6 


1 


2672 


2679 


2685 


2692 


2698 


6 


2 


2737 


2743 


2750 


2756 


2763 


6 


3 


2802 


2808 


2814 


2821 


2827 


6 


4 


2866 


2872 


2879 


2885 


2892 


6 


6 


82930 


82937 


82943 


82950 


82956 


6 


6 


2995 


3001 


3008 


3014 


3020 


6 


7 


3059 


3065 


3072 


3078 


3085 


6 


8 


3123 


3129 


3136 


3142 


3149 


6 


9 


3187 


3193 


3200 


3206 


3213 


6 


680 


83251 


83257 


83264 


83270 


83276 


6 


1 


3315 


3321 


3327 


3334 


3340 


6 


2 


3378 


3385 


3391 


3398 


3404 


6 


3 


3442 


3448 


3455 


3461 


3467 


6 


4 


3506 


3512 


3518 


3525 


3531 


6 


5 


83569 


83575 


83582 


83588 


83594 


6 


6 


3632 


3639 


3645 


3651 


3658 


6 


7 


3696 


3702 


3708 


3715 


3721 


6 


8 


3759 


3765 


3771 


3778 


3784 


6 


9 


3822 


3828 


3835 


3841 


3847 


6 


690 


83885 


83891 


83897 


83904 


83910 


6 


1 


3948 


3954 


3960 


3967 


3973 


6 


2 


4011 


4017 


4023 


4029 


4036 


6 


3 


4073 


4080 


4086 


4092 


4098 


6 


4 


4136 


4142 


4148 


4155 


4161 


6 


6 


84198 


84205 


84211 


84217 


84223 


6 


6 


4261 


4267 


4273 


4280 


4286 


6 


7 


4323 


4330 


4336 


4342 


4348 


6 


8 


4386 


4392 


4398 


4404 


4410 


6 


9 


4448 


4454 


4460 


4466 


4473 


6 



174 



Of NUMBER^ 



N 


5 


6 


7 


8 


9 


D 


640 


80652 


80659 


80665 


80672 


80679 


7 


1 


0720 


0726 


0733 


0740 


0747 


7 


2 


0787 


0794 


0801 


0808 


0814 


7 


3 


0855 


0862 


0868 


0875 


0882 


7 


4 


0922 


0929 


0936 


0943 


0949 


7 


5 


80990 


80996 


81003 


81010 


81017 


7 


6 


1057 


1064 


1070 


1077 


1084 


7 


7 


1124 


1131 


1137 


1144 


1151 


7 


8 


1191 


1198 


1204 


1211 


1218 


7 


9 


1258 


1265 


1271 


1278 


1285 


7 


650 


81325 


81331 


81338 


81345 


81351 


7 


1 


1391 


1398 


1405 


1411 


1418 


7 


3 


1458 


1465 


1471 


1478 


1485 


7 


3 


1525 


1531 


1538 


1544 


1551 


7 


4 


1591 


1598 


1604 


1611 


1617 


7 


5 


81657 


81664 


81671 


81677 


81684 


7 


6 


1723 


1730 


1737 


1743 


1750 


7 


7 


1790 


1796 


1803 


1809 


1816 


7 


8 


1856 


1862 


1869 


1875 


1882 


7 


9 


1921 


1928 


1935 


1941 


1948 


7 


660 


81987 


81994 


82000 


82007 


82014 


7 


1 


2053 


2060 


2066 


2073 


2079 


7 


2 


2119 


2125 


2132 


2138 


2145 


7 


3 


2184 


2191 


2197 


2204 


2210 


7 


4 


2249 


2256 


2263 


2269 


2276 


7 


5 


82315 


82321 


82328 


82334 


82341 


7 


6 


2380 


2387 


2393 


2400 


2406 


7 


7 


2445 


2452 


2458 


2465 


2471 


6 


8 


2510 


2517 


2523 


2530 


2536 


6 


9 


2575 


2582 


2588 


2595 


2601 


6 


670 


82640 


82646 


82653 


82659 


82666 


6 


1 


2705 


2711 


2718 


2724 


2730 


6 


3 


2769 


2776 


2782 


2789 


2795 


6 


3 


2834 


2840 


2847 


2853 


2860 


6 


4 


2898 


2905 


2911 


2918 


2924 


6 


5 


82963 


82969 


82975 


82982 


82988 


6 


6 


3027 


3033 


3040 


3046 


3052 


6 


7 


3091 


3097 


3104 


3110 


3117 


6 


8 


3155 


3161 


3168 


3174 


3181 


6 


9 


3219 


3225 


3232 


3238 


3245 


6 


680 


83283 


83289 


83296 


83302 


83308 


6 


1 


3347 


3353 


3359 


3366 


3372 


6 


2 


3410 


3417 


3423 


3429 


3436 


6 


3 


3474 


3480 


3487 


3493 


3499 


6 


4 


3537 


•3544 


3550 


3556 


3563 


6 


5 


83601 


83607 


83613 


83620 


83620 


6 


6 


3664 


3670 


3677 


3683 


3689 


6 


7 


3727 


3734 


3740 


3746 


3753 


6 


8 


3790 


3797 


3803 


3809 


3816 


6 


9 


3853 


3860 


3866 


3872 


3879 


6 


690 


83916 


83923 


83929 


83935 


83942 


6 


1 


3979 


3985 


3992 


3998 


4004 


6 


2 


4042 


• 4048 


4055 


4061 


4067 


6 


3 


4105 


4111 


4117 


4123 


4130 


6 


4 


4167 


4173 


4180 


4186 


4192 


6 


5 


84230 


84236 


84242 


84248 


84255 


6 


6 


4292 


4298 


4305 


4311 


4317 


6 


7 


4354 


4361 


4367 


4373 


4379 


6 


8 


4417 


4423 


4429 


4435 


4442 


6 


9 


4479 


4485 


4491 


4497 


4504 


6 



175 



I. LOGARITHMS 



N 





1 


2 


3 


4 


D 


700 


84510 


84516 


84522 


84528 


84535 


6 


1 


4572 


4578 


4584 


4590 


4597 


6 


2 


4634 


4640 


4646 


4652 


4658 


6 


3 


4696 


4702 


4708 


4714 


4720 


6 


4 


4757 


4763 


4770 


4776 


4782 


6 


5 


84819 


84825 


84831 


84837 


84844 


6 


6 


4880 


4887 


4893 


4899 


4905 


6 


7 


4942 


4948 


4954 


4960 


4967 


6 


8 


5003 


5009 


5016 


5022 


5028 


6 


9 


5065 


5071 


5077 


5083 


5089 


6 


710 


85126 


85132 


85138 


85144 


85150 


6 


1 


5187 


5193 


5199 


5205 


5211 


6 


2 


5248 


5254 


5260 


5266 


5272 


6 


3 


5309 


5315 


5321 


5327 


5333 


6 


4 


5370 


5376 


5382 


5388 


5394 


6 


6 


85431 


85437 


85443 


85449 


85455 


6 


6 


5491 


5497 


5503 


5509 


5516 


6 


7 


5552 


5558 


5564 


5570 


5576 


6 


8 


5612 


5618 


5625 


5631 


5637 


6 


9 


5673 


5679 


5685 


5691 


5697 


6 


730 


85733 


85739 


85745 


85751 


85757 


6 


1 


5794 


5800 


5806 


5812 


5818 


6 


2 


5854 


5860 


5866 


5872 


5878 


6 


3 


5914 


5920 


5926 


5932 


5938 


6 


4 


5974 


5980 


5986 


5992 


5998 


6 


5 


86034 


86040 


86046 


86052 


86058 


6 


6 


6094 


6100 


6106 


6112 


6118 


6 


7 


6153 


6159 


6165 


6171 


6177 


6 


8 


6213 


6219 


6225 


6231 


6237 


6 


9 


6273 


6279 


6285 


6291 


6297 


6 


730 


86332 


86338 


86344 


86350 


86356 


6 


1 


6392 


6398 


6404 


6410 


6415 


6 


2 


6451 


6457 


6463 


6469 


6475 


6 


3 


6510 


6516 


6522 


6528 


6534 


6 


4 


6570 


6576 


6581 


6587 


6593 


6 


5 


86629 


86635 


86641 


86646 


86652 


6 


6 


6688 


6694 


6700 


6705 


6711 


6 


7 


6747 


6753 


6759 


6764 


6770 


6 


8 


6806 


6812 


6817 


6823 


6829 


6 


9 


6864 


6870 


6876 


6882 


6888 


6 


740 


86923 


86929 


86935 


86941 


86947 


6 


1 


6982 


6988 


6994 


6999 


7005 


6 


2 


7040 


7046 


7052 


7058 


7064 


6 


3 


7099 


7105 


7111 


7116 


7122 


6 


4 


7157 


7163 


7169 


7175 


7181 


6 


5 


87216 


87221 


87227 


87233 


87239 


6 


6 


7274 


7280 


7286 


7291 


7297 


6 


7 


7332 


7338 


7344 


7349 


7355 


6 


8 


7390 


7396 


7402 


7408 


7413 


6 


9 


7448 


7454 


7460 


7466 


7471 


6 


750 


87506 


87512 


87518 


87523 


87529 


6 


1 


7564 


7570 


757G 


7581 


7587 


6 


2 


7622 


7628 


7633 


7639 


7645 


6 


3 


7679 


7685 


7691 


7697 


7703 


6 


4 


7737 


7743 


7749 


7754 


7760 


6 


6 


87795 


87800 


87806 


87812 


87818 


6 


6 


7852 


7858 


7864 


7869 


7875 


6 


7 


7910 


7915 


7921 


7927 


7933 


6 


8 


7967 


7973 


7978 


7984 


7990 


6 


9 


8024 


8030 


8036 


8041 


8047 


« 



176 



OF NUMBERS 



^ 



N 


5 


6 


7 


8 


9 


D 


700 


84541 


84547 


64553 


84559 


84566 


6 


1 


4603 


4609 


4615 


4621 


4628 


6 


2 


4665 


4671 


4677 


4683 


4689 


6 


a 


4726 


4733 


4739 


4745 


4751 


6 


4 


4788 


4794 


4800 


4807 


4813 


6 


5 


84850 


84856 


84862 


84868 


84874 


6 


6 


4911 


4917 


4924 


4930 


4936 


6 


7 


4973 


4979 


4985 


4991 


4997 


6 


8 


5034 


5040 


5046 


5052 


5058 


6 


9 


5095 


5101 


5107 


5114 


5120 


6 


710 


85156 


85163 


85169 


85175 


85181 


6 


1 


5217 


5224 


5230 


5236 


5242 


6 


2 


5278 


5285 


5291 


5297 


5303 


6 


3 


5339 


5345 


5352 


5358 


5364 


6 


4 


5400 


5406 


5412 


5418 


5425 


6 


5 


85461 


85467 


85473 


85479 


85485 


6 


6 


5522 


5528 


5534 


5540 


5546 


6 


7 


5582 


5588 


5594 


5600 


5606 


6 


8 


5643 


5649 


5655 


5661 


5667 


6 


9 


5703 


5709 


5715 


5721 


5727 


6 


720 


85763 


85769 


85775 


85781 


85788 


6 


1 


5824 


5830 


5836 


5842 


5848 


6 


2 


5884 


5890 


5896 


5902 


5908 


6 


3 


5944 


5950 


5956 


5962 


5968 


6 


4 


6004 


6010 


6016 


6022 


6028 


6 


5 


86064 


86070 


86076 


86082 


86088 


6 


6 


6124 


6130 


6136 


6141 


6147 


6 


7 


6183 


6189 


6195 


6201 


6207 


6 


8 


6243 


6249 


6255 


6261 


6267 


6 


9 


6303 


6308 


6314 


6320 


6326 




730 


86362 


86368 


86374 


86380 


86386 


6 


1 


6421 


6427 


6433 


6439 


6445 


6 


2 


6481 


6487 


6493 


6499 


6504 


6 


3 


6540 


6546 


6552 


6558 


6564 


6 


4 


6599 


6605 


6611 


6617 


6623 


6 


5 


86658 


86646 


86670 


86676 


86682 


6 


6 


6717 


6723 


6729 


6735 


6741 


6 


7 


6776 


6782 


6788 


6794 


6800 


6 


8 


6835 


6841 


6847 


6853 


6859 


6 


9 


6894 


6900 


6906 


6911 


6917 


6 


740 


86953 


86958 


86964 


86970 


86976 


6 


1 


7011 


7017 


7023 


7029 


7035 


6 


2 


7070 


7075 


7081 


7087 


7093 


6 


3 


7128 


7134 


7140 


7146 


7151 


6 


4 


7186 


7192 


7198 


7204 


7210 


6 


5 


87245 


87251 


87256 


87262 


87268 


6 


6 


7303 


7309 


7315 


7320 


7326 


6 


7 


7361 


7367 


7373 


7379 


7384 


6 


8 


7419 


7425 


7431 


7437 


7442 


6 


9 


7477 


7483 


7489 


7495 


7500 


6 


750 


87535 


87541 


87547 


87552 


87558 


6 


1 


7593 


7599 


7604 


7610 


7616 


6 


2 


7651 


7656 


7662 


7668 


7674 


6 


3 


7708 


7714 


7720 


7726 


7731 


6 


4 


7766 


7772 


7777 


7783 


7789 


6 


5 


87823 


87829 


87835 


87841 


87846 


6 


6 


7881 


7887 


7892 


7898 


7904 


6 


7 


7938 


7944 


7950 


7955 


7961 


6 


8 


7996 


8001 


8007 


8013 


8018 


6 


. 9 


8053 


8058 


8064 


8070 


8076 


6 



177 



1. LOGARITHMS 



N 





1 


2 


3 


4 


D 
~6 


760 


88081 


88087 


88093 


88098 


88104 


1 


8138 


8144 


8150 


8156 


8161 


6 


2 


8195 


8201 


8207 


8213 


8218 


6 


3 


8252 


8258 


8264 


8270 


8275 


6 


4 


8309 


8315 


8321 


8326 


8332 


6 


6 


88366 


88372 


88377 


88383 


88389 


6 


6 


8423 


8429 


8434 


8440 


8446 


6 


7 


8480 


8485 


8491 


8497 


8502 


6 


8 


8536 


8542 


8547 


8553 


8559 


6 


9 


8593 


8598 


8604 


8610 


8615 


6 


770 


88649 


88655 


88660 


88666 


88672 


6 


1 


8705 


8711 


8717 


8722 


8728 


6 


2 


8762 


8767 


8773 


8779 


8784 


6 


3 


8818 


8824 


8829 


8835 


8840 


6 


4 


8874 


8880 


8885 


8891 


8897 


6 


5 


88930 


88936 


88941 


88947 


88953 


6 


6 


8986 


8992 


8997 


9003 


9009 


6 


7 


9042 


9048 


9053 


9059 


9064 


6 


8 


9098 


9104 


9109 


9115 


9120 


6 


9 


9154 


9159 


9165 


9170 


9176 


6 


780 


89209 


89215 


89221 


89226 


89232 


6 


1 


9265 


9271 


9276 


9282 


9287 


6 


2 


9321 


9326 


9332 


9337 


9343 


6 


3 


9376 


9382 


9387 


9393 


9398 


6 


4 


9432 


9437 


9443 


9448 


9454 


6 


5 


89487 


89492 


89498 


89504 


89509 


6 


6 


9542 


9548 


9553 


9559 


9564 


5 


7 


9597 


9603 


9609 


9614 


9620 


5 


8 


9653 


9658 


9664 


9669 


9675 


5 


9 


9708 


9713 


9719 


9724 


9730 


5 


790 


89763 


89768 


89774 


89779 


89785 


5 


1 


9818 


9823 


9829 


9834 


9840 


5 


2 


9873 


9878 




9889 


9894 


5 


3 


9927 


9933 


9938 


9944 


9949 


5 


4 


9982 


9988 


9993 


9998 


90004 


5 


5 


90037 


90042 


90048 


90053 


90059 


5 


6 


0091 


0097 


0102 


OlOS 


0113 


5 


7 


0146 


0151 


0157 


0162 


0168 


5 


8 


0200 


0206 


0211 


0217 


0222 


5 


9 


0255 


0260 


0266 


0271 


0276 


5 


800 


90309 


90314 


90320 


90325 


90331 


6 


1 


0363 


0369 


0374 


0380 


0385' 


5 


2 


0417 


0423 


0428 


0434 


0439 


6 


3 


0472 


0477 


0482 


0488 


0493 


5 


4 


0526 


0531 


0536 


0542 


0547 


5 


6 


90580 


90585 


90590 


90596 


90601 


6 


6 


0634 


0639 


0644 


0650 


0655 


5 


7 


0687 


0693 


0698 


0703 


0709 


5 


8 


0741 


0747 


0752 


0757 


0763 


5 


9 


0795 


0800 


0806 


0811 


0816 


5 


810 


90849 


90854 


90859 


90865 


90870 


5 


1 


0902 


0907 


0913 


0918 


0924 


5 


2 


0956 


0961 


0966 


0972 


0977 


5 


3 


1009 


1014 


1020 


1025 


1030 


5 


4 


1062 


1068 


1073 


1078 


1084 


5 


5 


91116 


91121 


91126 


91132 


91137 


5 


6 


1169 


1174 


1180 


1185 


1190 


5 


7 


1222 


1228 


1233 


1238 


1243 


5 


8 


1275 


1281 


1286 


1291 


1297 


5 


9 


1328 


1334 


1339 


1344 


1350 


6 



178 



OF NUMBERS 



N 


5 


6 


7 


8 


9 


D 


760 


88110 


88116 


88121 


88127 


88133 


6 


1 


8167 


8173 


8178 


8184 


8190 


6 


2 


8224 


8230 


8235 


8241 


8247 


6 


3 


8281 


8287 


8292 


8298 


8304 


6 


4 


8338 


8343 


8349 


8355 


8360 


6 


5 


88395 


88400 


88406 


88412 


88417 


6 


6 


8451 


8457 


8463 


8468 


8474 


6 


7 


8508 


8513 


8519 


8525 


8530 


6 


8 


8564 


8570 


8576 


8581 


8587 


6 


9 


8621 


8627 


8632 


8638 


8643 


6 


770 


88677 


88683 


88689 


88694 


88700 


6 


1 


8734 


8739 


8745 


8750 


8756 


6 


2 


8790 


8795 


8801 


8807 


8812 


6 


3 


8846 


8852 


8857 


8863 


8868 


6 


4 


8902 


8908 


8913 


8919 


8925 


6 


5 


88958 


88964 


88969 


88975 


88981 


6 


6 


9014 


9020 


9025 


9031 


9037 


6 


7 


9070 


9076 


9081 


9087 


9092 


6 


8 


9126 


9131 


9137 


9143 


9148 


6 


9 


9182 


9187 


9193 


9198 


9204 


6 


780 


89237 


89243 


89248 


89254 


89260 


6 


1 


9293 


9298 


9304 


9310 


9315 


6 


2 


9348 


9354 


9360 


9365 


9371 


6 


3 


9404 


9409 


9415 


9421 


9426 


6 


4 


9459 


9465 


9470 


9476 


9481 


6 


5 


89515 


89520 


89526 


89531 


89537 


6 


6 


9570 


9575 


9581 


9586 


9592 


5 


7 


9625 


9631 


9636 


9642 


9647 


5 


8 


9680 


9686 


9691 


9697 


9702 


5 


9 


9735 


9741 


9746 


9752 


9757 


5 


790 


89790 


89796 


89801 


89807 


89812 


5 


1 


9845 


9851 


9856 


9862 


9867 


5 


2 


9900 


9905 


9911 


9916 


9922 


5 


3 


9955 


9960 


9966 


9971 


9977 


5 


4 


90009 


90015 


90020 


90026 


90031 


5 


5 


90064 


90069 


90075 


90080 


90086 


5 


6 


0119 


0124 


0129 


0135 


0140 


5 


7 


0173 


0179 


0184 


0189 


0195 


5 


8 


0227 


0233 


0238 


0244 


0249 


5 


9 


0282 


0287 


0293 


0298 


0304 


5 


800 


90336 


90342 


90347 


90352 


90358 


5 


1 


0390 


0396 


0401 


0407 


0412 


5 


2 


0445 


0450 


0455 


0461 


0466 


5 


3 


0499 


0504 


0509 


0515 


0520 


5 


4 


0553 


0558 


0563 


0569 


0574 


5 


5 


90607 


90612 


90617 


90623 


90628 


5 


6 


0660 


0666 


0671 


0677 


0682 


5 


7 


0714 


0720 


0725 


0730 


0736 


5 


8 


0768 


0773 


0779 


07X4 


0789 


5 


9 


0822 


0827 


0832 


0838 


0843 


5 


810 


90875 


90881 


90886 


90891 


90897 


5 


1 


0929 


0934 


0940 


0945 


0950 


5 


2 


0982 


0988 


0993 


0998 


1004 


5 


3 


1036 


1041 


1046 


1052 


1057 


5 


4 


1089 


1094 


1100 


1105 


1110 


5 


5 


91142 


91148 


91153 


91158 


91164 


5 


6 


1196 


1201 


1206 


1212 


1217 


5 


7 


1249 


1254 


1259 


1265 


1270 


5 


8 


1302 


1307 


1312 


1318 


1323 


5 


9 


1355 


1360 


1365 


1371 


1376 


S 



179 



I. LOGARITHMS 



N 





1 


3 


3 


4 


D 


830 


91381 


91387 


91392 


91397 


91403 


5 


1 


1434 


1440 


1445 


1450 


1455 


5 


2 


1487 


1492 


1498 


1503 


1508 


5 


3 


1540 


1545 


1551 


1556 


1561 


5 


4 


1593 


1598 


1603 


1609 


1614 


5 


5 


91645 


91651 


91656 


91661 


91666 


5 


6 


1698 


1703 


1709 


1714 


1719 


5 


7 


1751 


1756 


1761 


1766 


1772 


5 


8 


1803 


1808 


1814 


1819 


1824 


5 


9 


1855 


1861 


1866 


1871 


1876 


5 


830 


91908 


91913 


91918 


91924 


91929 


5 


1 


1960 


1965 


1971 


1976 


1981 


5 


3 


2012 


2018 


2023 


2028 


2033 


5 


3 


2065 


2070 


2075 


2080 


2085 


5 


4 


2117 


2122 


2127 


2132 


2137 


5 


5 


92169 


92174 


92179 


92184 


92189 


5 


6 


2221 


2226 


2231 


2236 


2241 


5 


7 


2273 


2278 


2283 


2288 


2293 


5 


8 


2324 


2330 


2335 


2340 


2345 


5 


9 


2376 


2381 


2387 


2392 


2397 


5 


840 


92428 


92433 


92438 


92443 


92449 


5 


1 


2480 


2485 


2490 


2495 


2500 


5 


3 


2531 


2536 


2542 


2547 


2552 


5 


3 


2583 


2588 


2593 


2598 


2603 


5 


4 


2634 


2639 


2645 


2650 


2655 


5 


5 


92686 


92691 


92696 


92701 


92706 


5 


6 


2737 


2742 


2747 


2752 


2758 


5 


7 


2788 


2793 


2799 


2804 


2809 


5 


8 


2840 


2845 


2850 


2855 


2860 


5 


9 


2891 


2896 


2901 


2906 


2911 


5 


850 


92942 


92947 


92952 


92957 


92962 


5 


1 


2993 


2998 


3003 


3008 


3013 


5 


2 


3044 


3049 


3054 


3059 


3064 


5 


3 


3095 


3100 


3105 


3110 


3115 


5 


4 


3146 


3151 


3156 


3161 


3166 


5 


5 


93197 


93202 


93207 


93212 


93217 


5 


6 


3247 


3252 


3258 


3263 


3268 


5 


7 


3298 


3303 


3308 


3313 


3318 


5 


8 


3349 


3354 


3359 


3364 


3369 


5 


9 


3399 


3404 


3409 


3414 


3420 


5 


860 


93450 


93455 


93460 


93465 


93470 


5 


1 


3500 


3505 


3510 


3515 


3520 


5 


3 


3551 


3556 


3561 


3566 


3571 


5 


3 


3601 


3606 


3611 


3616 


3621 


5 


4 


3651 


3656 


3661 


3666 


3671 


5 


5 


93702 


93707 


93712 


93717 


93722 


5 


6 


3752 


3757 


3762 


3767 


3772 


5 


7 


3802 


3807 


3812 


3817 


3822 


5 


8 


3852 


3857 


3862 


3867 


3872 


o 


9 


3902 


3907 


3912 


3917 


3922 


5 


^70 


93952 


93957 


93962 


93967 


93972 


5 


1 


4002 


4007 


4012 


4017 


4022 


5 


3 


4052 


4057 


40C2 


4067 


4072 


5 


3 


4101 


4106 


4111 


4116 


4121 


5 


4 


4151 


4156 


4161 


4166 


4171 


5 


5 


94201 


94206 


94211 


94216 


94221 


5 


6 


4250 


4255 


4260 


4265 


4270 


5 


7 


4300 


4305 


4310 


4315 


4320 


5 


8 


4349 


4354 


4359 


4364 


4369 


5 


9 


4399 


4404 


4409 


4414 


4419 


5 



180 



OF NUMBERS 



N 


5 


6 


7 


8 


9 

91429 


D 


830 


91408 


91413 


91418 


91424 


5 


1 


1461 


1466 


1471 


1477 


1482 


5 


2 


1514 


1519 


1524 


1529 


1535 


5 


3 


1566 


1572 


1577 


1582 


1587 


5 


4 


1619 


1624 


1630 


1635 


1640 


5 


5 


91672 


91677 


91682 


91687 


91693 


5 


6 


1724 


1730 


1735 


1740 


1745 


5 


7 


1777 


1782 


1787 


1793 


1798 


5 


8 


1829 


1834 


1840 


1845 


1850 


5 


9 


1882 


1887 


1892 


1897 


1903 


5 


830 


91934 


91939 


91944 


91950 


91955 


5 


1 


1986 


1991 


1997 


2002 


2007 


5 


2 


2038 


2044 


2049 


2054 


2059 


5 


3 


2091 


2096 


2101 


2106 


2111 


5 


4 


2143 


2148 


2153 


2158 


2163 


5 


5 


92195 


92200 


92205 


92210 


92215 


5 


6 


2247 


2252 


2257 


2262 


2267 


5 


7 


2298 


2304 


2309 


2314 


2319 


5 


8 


2350 


2355 


2361 


2366 


2371 


5 


9 


2402 


2407 


2412 


2418 


2423 


5 


840 


92454 


92459 


92464 


92469 


92474 


5 


1 


2505 


2511 


2516 


2521 


2526 


5 


2 


2557 


2562 


2567 


2572 


2578 


5 


3 


2609 


2614 


2619 


2624 


2629 


5 


4 


2660 


2665 


2670 


2675 


2681 


5 


5 


92711 


92716 


92722 


92727 


92732 


5 


6 


2763 


2768 


2773 


2778 


2783 


5 


7 


2814 


2819 


2824 


2829 


2834 


5 


8 


2865 


2870 


2875 


2881 


2886 


5 


9 


2916 


2921 


2927 


2932 


2937 


5 


850 


92967 


92973 


92978 


92983 


92988 


5 


1 


3018 


3024 


3029 


3034 


3039 


5 


2 


3069 


3075 


3080 


3085 


3090 


5 


3 


3120 


3125 


3131 


3136 


3141 


5 


4 


3171 


3176 


3181 


3186 


3192 


5 


5 


93222 


93227 


93232 


93237 


93242 


5 


6 


3273 


3278 


3283 


3288 


3293 


5 


7 


3323 


3328 


3334 


3339 


3344 


5 


8 


3374 


3379 


3384 


3389 


3394 


5 


9 


3425 


3430 


3435 


3440 


3445 


5 


860 


93475 


93480 


93485 


93490 


93495 


5 


1 


3526 


3531 


3536 


3541 


3546 


5 


2 


3576 


3581 


3586 


3591 


3596 


5 


3 


3626 


3631 


3636 


3641 


3646 


5 


4 


3676 


3682 


3687 


3692 


3697 


5 


5 


93727 


93732 


93737 


93742 


93747 


5 


6 


3777 


3782 


3787 


3792 


3797 


5 


7 


3827 


3832 


3837 


3842 


3847 


5 


8 


3877 


3882 


3887 


3892 


3897 


5 


9 


3927 


3932 


3937 


3942 


3947 


5 


870 


93977 


93982 


93987 


93992 


93997 


5 


1 


4027 


4032 


4037 


4042 


4047 


5 


*> 


4077 


4082 


4086 


4091 


4096 


5 


3 


4126 


4131 


4136 


4141 


4146 


5 


4 


4176 


4181 


4186 


4191 


4196 


5 


5 


94226 


94231 


94236 


94240 


94245 


5 


6 


4275 


4280 


4285 


4290 


4295 


5 


7 


4325 


4330 


4335 


4340 


4345 


5 


8 


4374 


4379 


4384 


4389 


4394 


5 


9 


4424 


4429 


4433 


4438 


4443 


5 



181 



I. LOGARITHMS 



N 





1 


2 


3 


4 


D 
5 


880 


94448 


94453 


94458 


94463 


94468 


1 


4498 


4503 


4507 


4512 


4517 


5 


2 


4547 


4552 


4557 


4562 


4567 


5 


3 


4596 


4601 


4606 


4611 


4616 


5 


4 


4645 


4650 


4655 


4660 


4665 


5 


5 


94694 


94699 


94704 


94709 


94714 


5 


6 


4743 


4748 


4753 


4758 


4763 


5 


7 


4792 


4797 


4802 


4807 


4812 


5 


8 


4841 


4846 


4851 


4856 


4861 


5 


9 


4890 


4895 


4900 


4905 


4910 


5 


890 


94939 


94944 


94949 


94954 


94959 


5 


1 


4988 


4993 


4998 


5002 


5007 


5 


2 


5036 


5041 


5046 


5051 


5056 


5 


3 


5085 


5090 


5095 


5100 


5105 


5 


4 


5134 


5139 


5143 


5148 


5153 


5 


5 


95182 


95187 


95192 


95197 


95202 


5 


6 


5231 


5236 


5240 


5245 


5250 


5 


7 


5279 


5284 


5289 


5294 


5299 


5 


8 


5328 


5332 


5337 


5342 


5347 


5 


9 


5376 


5381 


5386 


5390 


5395 


5 


900 


95424 


95429 


95434 


95439 


95444 


5 


1 


5472 


5477 


5482 


5487 


5492 


5 


2 


5521 


5525 


5530 


5535 


5540 


5 


3 


5569 


5574 


5578 


5583 


5588 


5 


4 


5617 


5622 


5626 


5631 


5636 


5 


5 


95665 


95670 


95674 


95679 


95684 


5 


6 


5713 


5718 


5722 


5727 


5732 


5 


7 


5761 


5766 


5770 


5775 


5780 


5 


8 


5809 


5813 


5818 


5823 


5828 


5 


9 


5856 


5861 


5866 


5871 


5875 


5 


910 


95904 


95909 


95914 


95918 


95923 


5 


1 


5952 


5957 


5961 


5960 


5971 


5 


2 


5999 


6004 


6009 


6014 


6019 


5 


3 


6047 


6052 


6057 


6061 


6066 


5 


4 


6095 


6099 


6104 


6109 


6114 


5 


5 


96142 


96147 


96152 


96156 


96161 


5 


6 


6190 


6194 


6199 


6204 


6209 


5 


7 


6237 


6242 


6246 


6251 


6256 


5 


8 


6284 


6289 


6294 


6298 


6303 


5 


9 


6332 


6336 


6341 


6346 


6350 


5 


920 


96379 


96384 


96388 


96393 


96398 


5 


1 


6426 


6431 


6435 


6440 


6445 


5 


2 


6473 


6478 


6483 


6487 


6492 


5 


3 


6520 


6525 


6530 


6534 


6539 


5 


4 


6567 


6572 


6577 


6581 


6586 


5 


5 


96614 


96619 


96624 


96628 


96633 


5 


3 


6661 


6666 


6670 


6675 


6680 


5 


7 


6708 


6713 


6717 


6722 


6727 


5 


8 


6755 


6759 


6764 


6769 


6774 


5 


9 


6802 


6806 


6811 


6816 


6820 


5 


930 


96848 


96853 


96858 


96862 


96867 


5 


1 


6895 


6900 


6904 


6909 


6914 


5 


2 


6942 


6946 


6951 


6956 


6960 


5 


3 


6988 


6993 


6997 


7002 


7007 


5 


4 


7035 


7039 


7044 


7049 


7053 


5 


5 


97081 


97086 


97090 


97095 


97100 


5 


6 


7128 


7132 


7137 


7142 


7146 


5 


7 


7174 


7179 


7183 


7188 


7192 


5 


8 


7220 


7225 


7230 


7234 


7239 


5 


9 


7267 


7271 


7276 


7280 


7285 


5 



182 



OF NUMBERS 



N 


5 


6 


7 


8 


9 


D 


880 


94473 


94478 


94483 


94488 


94493 


5 


1 


4522 


4527 


4532 


4537 


4542 


5 


2 


4571 


4576 


4581 


4586 


4591 


5 


3 


4621 


4626 


4630 


4635 


4640 


5 


4 


4670 


4675 


4680 


4685 


4689 


5 


5 


94719 


94724 


94729 


94734 


94738 


5 


6 


4768 


4773 


4778 


4783 


4787 


5 


7 


4817 


4822 


4827 


4832 


4836 


5 


8 


4866 


4871 


4876 


4880 


4885 


5 


9 


4915 


4919 


4924 


4929 


4934 


5 


890 


94963 


94968 


94973 


94978 


94983 


5 


1 


5012 


5017 


5022 


5027 


5032 


5 


2 


5061 


5066 


5071 


5075 


5080 


5 


3 


5109 


5114 


5119 


5124 


5129 


5 


4 


5158 


5163 


5168 


5173 


5177 


5 


5 


95207 


95211 


95216 


95221 


95226 


5 


6 


5255 


5260 


5265 


5270 


5274 


5 


7 


5303 


5308 


5313 


5318 


5323 


5 


8 


5352 


5357 


5361 


5366 


5371 


5 


9 


5400 


5405 


5410 


5415 


5419 


5 


900 


95448 


95453 


95458 


95463 


95468 


5 


1 


5497 


5501 


5506 


5511 


5516 


5 


2 


5545 


5550 


5554 


5559 


5564 


5 


3 


5593 


5598 


5602 


5607 


5612 


5 


4 


5641 


5646 


5650 


5655 


5660 


5 


5 


95689 


95694 


95698 


95703 


95708 


5 


6 


5737 


5742 


5746 


5751 


5756 


5 


7 


5785 


5789 


5794 


5799 


5804 


5 


8 


5832 


5837 


5842 


5847 


5852 


5 


9 


5880 


5885 


5890 


5895 


5899 


5 


910 


95928 


95933 


95938 


95942 


95947 


5 


1 


5976 


5980 


5985 


5990 


5995 


5 


2 


6023 


6028 


6033 


6038 


6042 


5 


3 


6071 


6076 


6080 


6085 


6090 


5 


4 


6118 


6123 


6128 


6133 


6137 


5 


5 


96166 


96171 


96175 


96180 


96185 


5 


6 


6213 


6218 


6223 


6227 


6232 


5 


7 


6261 


6265 


6270 


6275 


6280 


5 


8 


6308 


6313 


6317 


6322 


6327 


5 


9 


6355 


6360 


6365 


6369 


6374 


5 


920 


96402 


96407 


96412 


96417 


96421 


5 


1 


6450 


6454 


6459 


6464 


6468 


5 


2 


6497 


6501 


6506 


6511 


6515 


5 


3 


6544 


6548 


6553 


6558 


6562 


5 


4 


6591 


6595 


6600 


6605 


6609 


5 


6 


96638 


96642 


96647 


96652 


96656 


5 


6 


6685 


6689 


6694 


6699 


6703 


5 


7 


6731 


6736 


6741 


6745 


6750 


5 


8 


6778 


6783 


6788 


6792 


6797 


5 


9 


6825 


6830 


6834 


6839 


6844 


5 


930 


96872 


96876 


96881 


96886 


96890 


5 


1 


6918 


6923 


6928 


4932 


6937 


5 


2 


6965 


6970 


6974 


6979 


6984 


5 


3 


7011 


7016 


7021 


7025 


7030 


5 


4 


7058 


7063 


7067 


7072 


7077 


5 


5 


97104 


97109 


97114 


97118 


97123 


5 


6 


7151 


7155 


7160 


7165 


7169 


5 


7 


7197 


7202 


7206 


7211 


7216 


5 


8 


7243 


7248 


7253 


7257 


7262 


5 


9 


7290 


7294 


7299 


7304 


7308 


5 



183 



I. LOGARITHMS 



N 





1 


2 


3 


4 


D 


940 


97313 


97317 


97322 


97327 


97331 


~5 


1 


7359 


7364 


7368 


7373 


7377 


5 


2 


7405 


7410 


7414 


7419 


7424 


5 


3 


7451 


7456 


7460 


7465 


7470 


5 


4 


7497 


7502 


7506 


7511 


7516 


5 


5 


97543 


97548 


97552 


97557 


97562 


5 


6 


7589 


7594 


7598 


7603 


7607 


5 


7 


7635 


7640 


7644 


7649 


7653 


5 


8 


7681 


7685 


7690 


7695 


7699 


5 


9 


7727 


7731 


7736 


7740 


7745 


5 


950 


97772 


97777 


97782 


97786 


97791 


5 


1 


7818 


7823 


7827 


7832 


7836 


5 


2 


7864 


7868 


7873 


7877 


7882 


5 


3 


7909 


7914 


7918 


7923 


7928 


5 


4 


7955 


7959 


7964 


7968 


7973 


5 


5 


98000 


98005 


98009 


98014 


98019 


6 


6 


8046 


8050 


8055 


8059 


8064 


5 


7 


8091 


8096 


8100 


8105 


8109 


5 


8 


8137 


8141 


8146 


8150 


8155 


5 


9 


8182 


8186 


8191 


8195 


8200 


5 


960 


98227 


98232 


98236 


98241 


98245 


5 


1 


8272 


8277 


8281 


8286 


8290 


5 


2 


8318 


8322 


8327 


8331 


8336 


4 


3 


8363 


8367 


8372 


8376 


8381 


4 


4 


8408 


8412 


8417 


8421 


8426 


4 


5 


98453 


98457 


98462 


98466 


98471 


4 


6 


8498 


8502 


8507 


8511 


8516 


4 


7 


8543 


8547 


8552 


8556 


8561 


4 


8 


8588 


8592 


8597 


8601 


8605 


4 


9 


8632 


8637 


8641 


8646 


8650 


4 


970 


98677 


98682 


98686 


98691 


98695 


4 


1 


8722 


8726 


8731 


8735 


8740 


4 


2 


8767 


8771 


8776 


8780 


8784 


4 


3 


8811 


8816 


8820 


8825 


8829 


4 


4 


8856 


8860 


8865 


8869 


8874 


4 


5 


98900 


98905 


98909 


98914 


98918 


4 


6 


8945 


8949 


8954 


8958 


8963 


4 


7 


8989 


8994 


8998 


9003 


9007 


4 


8 


9034 


9038 


9043 


9047 


9052 


4 


9 


9078 


9083 


9087 


9092 


9096 


4 


980 


.99123 


99127 


99131 


99136 


99140 


4 


1 


9167 


9171 


9176 


9180 


9185 


4 


2 


9211 


9216 


9220 


9224 


9229 


4 


3 


9255 


9260 


9264 


9269 


9273 


4 


4 


9300 


9304 


9308 


9313 


9317 


4 


5 


99344 


99348 


99352 


99357 


99361 


4 


6 


9388 


9392 


9396 


9401 


9405 


4 


7 


9432 


9436 


9441 


9445 


9449 


4 


8 


9476 


9480 


9484 


9489 


9493 


4 


9 


9520 


9524 


9528 


9533 


9537 


4 


990 


99564 


99568 


99572 


99577 


99581 


4 


1 


9607 


9612 


9616 


9621 


9625 


4 


2 


9651 


9656 


9660 


9664 


9669 


4 


3 


9695 


9699 


9704 


9708 


9712 


4 


4 


9739 


9743 


9747 


9752 


9756 


4 





99782 


99787 


99791 


99795 


99800 


4 




9826 


9830 


9835 


9839 


9843 


4 


7 


9870 


9874 


9878 


9883 


98S7 


4 


g 


9913 


9917 


9922 


9926 


9930 


4 


9 


9957 


9961 


9965 


9970 


9974 


4 


1000 


00000 


00004 


00009 


00013 


00017 


4 



184 



OF NUMBERS 



N 


5 


6 


7 


8 


9 


D 


940 


97336 


97340 


97345 


97350 


97354 


5 


1 


7382 


7387 


7391 


7396 


7400 


5 


2 


7428 


7433 


7437 


7442 


7447 


5 


3 


7474 


7479 


7483 


7488 


7493 


5 


4 


7520 


7525 


7529 


7534 


7539 


5 


5 


97566 


97571 


97575 


97580 


97585 


5 


6 


7612 


7617 


7621 


7626 


7630 


5 


7 


7658 


7663 


7667 


7672 


7676 


5 


8 


7704 


7708 


7713 


7717 


7722 


5 


9 


7749 


7754 


7759 


7763 


7768 


5 


950 


97795 


97800 


97804 


97809 


97813 


5 


1 


7841 


7845 


7850 


7855 


7859 


5 


3 


7886 


7891 


7896 


7900 


7905 


5 


3 


7932 


7937 


7941 


7946 


7950 


5 


4 


7978 


7982 


7987 


7991 


7996 


5 


5 


98023 


98028 


98032 


98037 


98041 


5 


6 


8068 


8073 


8078 


8082 


8087 


5 


7 


8114 


8118 


8123 


8127 


8132 


5 


8 


8159 


8164 


8168 


8173 


8177 


5 


9 


8204 


8209 


8214 


8218 


8223 


5 


960 


98250 


98254 


98259 


98263 


98268 


5 


1 


8295 


8299 


8304 


8308 


8313 


5 


2 


8340 


8345 


8349 


8354 


8358 


4 


3 


8385 


8390 


8394 


8399 


8403 


4 


4 


8430 


8435 


8439 


8444 


8448 


4 


5 


98475 


98480 


98484 


98489 


98493 


4 


6 


8520 


8525 


8529 


8534 


8538 


4 


7 


8565 


8570 


8574 


8579 


8583 


4 


8 


8610 


8614 


8619 


8623 


8628 


4 


9 


8655 


8659 


8664 


8668 


8673 


4 


970 


98700 


98704 


98709 


98713 


98717 


4 


1 


8744 


8749 


8753 


8758 


8762 


4 


2 


8789 


8793 


8798 


8802 


8807 


4 


3 


8834 


8838 


8843 


8847 


8851 


4 


4 


8878 


8883 


8887 


8892 


8896 


4 


5 


98923 


98927 


98932 


98936 


98941 


4 


6 


8967 


8972 


8976 


8981 


8985 


4 


7 


9012 


9016 


9021 


9025 


9029 


4 


8 


9056 


9061 


9065 


9069 


9074 


4 


9 


9100 


9105 


9109 


9114 


9118 


4 


980 


99145 


99149 


99154 


99158 


99162 


4 


1 


9189 


9193 


9198 


9202 


9207 


4 


2 


9233 


9238 


9242 


9247 


9251 


4 


3 


9277 


9282 


9286 


9291 


9295 


4 


4 


9322 


9326 


9330 


9335 


9339 


4 


5 


99366 


99370 


99374 


99379 


99383 


4 


6 


9410 


9414 


9419 


9423 


9427 


4 


7 


9454 


9458 


9463 


9467 


94/1 


4 


8 


9498 


9502 


9506 


9511 


9515 


4 


9 


9542 


9546 


9550 


9555 


9559 


4 


990 


99585 


99590 


99594 


99599 


99603 


4 


1 


9629 


9634 


9638 


9642 


9647 


4 


2 


9673 


9677 


9682 


9686 


9691 


4 


3 


9717 


9721 


9726 


9730 


9734 


4 


4 


9760 


9765 


9769 


9774 


9778 


4 


5 


99804 


99808 


99813 


99817 


99822 


4 


6 


9848 


9852 


9856 


9861 


9865 


4 


7 


9891 


9896 


9900 


9904 


9909 


4 


8 


9935 


9939 


9944 


9948 


9952 


4 


1 9 


9978 


^9983 


9987 


9991 


9996 


4 


1 1000 


00022 


00026 


00030 


00035 


00039 


4 



185 



11. LOGARITHMIC 





179° 


178° 


( 177° 


176° 


175° 


Sin 


0° 


1° 


2° 


3° 


4° 


0' 


— X 


8.24186 


8.54282 


8.71880 


8.84358 


1 


6.46373 


24903 


64642 


72120 


84539 


2 


76476 


25609 


64999 


72359 


84718 


3 


94085 


26304 


65354 


72597 


84897 


4 


7.06579 


26988 


55705 


72834 


85075 


5 


7.16270 


8.27661 


8.66054 


8.73069 


8.85252 


6 


24188 


28324 


66400 


73303 


85429 


7 


30882 


28977 


66743 


73635 


85605 


8 


36682 


29621 


67084 


73767 


85780 


9 


41797 


30255 


67421 


73997 


85955 


10 


7.46373 


8.30879 


8.67757 


8.74226 


8.86128 


11 


50512 


31495 


58089 


74464 


86301 


13 


54291 


32103 


58419 


74680 


86474 


13 


57767 


32702 


58747 


74906 


86645 


14 


60985 


33292 


59072 


76130 


86816 


15 


7.63982 


8.33875 


8.59395 


8.76353 


8.86987 


16 


66784 


34450 


59715 


75675 


87156 


17 


69417 


35018 


60033 


75795 


87325 


18 


71900 


35578 


60349 


76015 


87494 


19 


74248 


36131 


60662 


76234 


87661 


20 


7.76476 


8.36678 


8.60973 


8.76451 


8.87829 


21 


78594 


37217 


61282 


76667 


87995 


22 


80615 


37750 


61589 


76883 


88161 


23 


82545 


38276 


61894 


77097 


88326 


24 


84393 


38796 


62196 


77310 


88490 


25 


7.86166 


8.39310 


8.62497 


8.77622 


8.88654 


26 


87870 


39818 


62795 


77733 


88817 


27 


89509 


40320 


63091 


77943 


88980 


28 


91088 


40816 


63385 


78152 


89142 


29 


92612 


41307 


63678 


78360 


89304 


30 


7.94084 


8.41792 


8.63968 


8.78568 


8.89464 


31 


95508 


42272 


64256 


78774 


89625 


33 


96887 


42746 


64643 


78979 


89784 


33 


98223 


43216 


64827 


79183 


89943 


34 


99520 


43680 


65110 


79386 


90102 


35 


8.00779 


8.44139 


8.65391 


8.79588 


8.90260 


36 


02002 


44594 


65670 


79789 


90417 


37 


03192 


45044 


65947 


79990 


90574 


38 


04350 


45489 


66223 


80189 


90730 


39 


05478 


45930 


66497 


80388 


90885 


40 


8.06578 


8.46366 


8.66769 


8.80585 


8.91040 


41 


07650 


46799 


67039 


80782 


91195 


43 


08696 


47226 


67308 


80978 


91349 


43 


09718 


47650 


67575 


81173 


91502 


44 


10717 


48069 


67841 


81367 


91655 


45 


8.11693 


8.48485 


8.68104 


8.81560 


8.91807 


46 


12647 


48896 


68367 


81762 


91969 


47 


13581 


49304 


68627 


81944 


92110 


48 


14495 


49708 


68886 


82134 


92261 


49 


15391 


50108 


69144 


82324 


92411 


50 


8. 16268 


8.50504 


8.69400 


8.82513 


8.92561 


51 


17128 


50897 


69654 


82701 


92710 


53 


17971 


51287 


69907 


82888 


92859 


53 


18798 


61673 


70159 


83075 


93007 


54 


19610 


52055 


70409 


83261 


93154 


55 


8.20407 


S. 62434 


8.70658 


8.83446 


8.93301 


56 


21189 


52810 


70905 


83630 


93448 


57 


21958 


53183 


71151 


83813 


93594 


58 


22713 


53552 


71395 


83996 


93740 


59 


23456 


63919 


71638 


84177 


93885 


60 


8.24186 


8 64282 


8. 71SS0 


8.84358 


8.94030 




89° 


88° 
91° 


87° 


86° 


85° 


Cos 


90° 


92° 


93° 


94° 



186 



SINES AND COSINES 



174^ 



8.94030 
94174 
94317 
94461 
94603 

8.94746 
94887 
95029 
95170 
95310 

8.95450 
95589 
95728. 
95867 
96005 

8.96143 
96280 
96417 
96553 
96689 

8.96825 
96960 
97095 
97229 
97363 

8.97496 
97629 • 
97762 
97894 
98026 

8.98157 
98288 
98419 
98549 
98679 

8.98808 
98937 
99066 
99194 
99322 

8.99450 
99577 
99704 
99830 
99956 

9.00082 
00207 
00332 
00456 
00581 

9.00704 
00828 
00951 
01074 
01196 

9.01318 
01440 
01561 
01682 
01803 

9.01923 

84° 
95<^ 



173' 



6° 



9.01923 
02043 
02163 
02283 
02402 

9.02520 
02639 
02757 
02874 
02992 

9.03109 
03226 
03342 
03458 
03574 

9.03690 
03805 
03920 
04034 
04149 

9.04262 
04376 
04490 
04603 
04715 

9.04828 
04940 
05052 
05164 
05275 

9.05386 
05497 
05607 
05717 
05827 

9.05937 
06046 
06155 
06264 
06372 

9.06481 
06589 
06696 
06804 
06911 

9.07018 
07124 
07231 
07337 
07442 

9.07548 
07653 
07758 
07863 
07968 

9.08072 
08176 
08280 
08383 
08486 

9.08589 

83° 
96° 



172° 


171° 


170° 1 


7° 


8° 


9° 


9.08589 


9.14356 


9.19433 


08692 


14445 


19513 


08795 


14535 


19592 


08897 


14624 


19672 


08999 


14714 


19751 


9.09101 


9.14803 


9.19830 


09202 


14891 


19909 


09304 


14980 


19988 


09405 


15069 


20067 


09506 


15157 


20145 


9.09606 


9.15245 


9.20223 


09707 


15333 


20302 


09807 


1.5421 


20380 


09907 


15508 


20468 


10006 


15596 


20535 


9.10106 


9.15683 


9.20613 


10205 


15770 


20691 


10304 


15857 


20768 


10402 


15944 


20845 


10501 


16030 


20922 


9.10599 


9.16116 


9.20999 


10697 


16203 


21076 


10795 


16289 


21153 


10893 


16374 


21229 


10990 


16460 


21306 


9.11087 


9.16545 


9.21382 


11184 


16631 


21458 


11281 


16716 


21534 


11377 


16801 


21610 


11474 


16886 


21685 


9.11570 


9.16970 


9.21761 


11666 


17055 


21836 


11761 


17139 


21912 


11857 


17223 


21987 


11952 


17307 


22062 


9.12047 


9.17391 


9.22137 


12142 


17474 


22211 


12236 


17558 


22286 


12331 


17641 


22361 


12425 


17724 


22435 


9.12519 


9.17807 


9.22509 


12612 


17890 


22583 


12706 


17973 


22657 


12799 


18055 


22731 


12892 


18137 


22805 


9.12985 


9.18220 


9.22878 


13078 


18302 


22952 


13171 


18383 


23025 


13263 


18465 


23098 


13355 


18547 


23171 


9.13447 


9.18628 


9.23244 


13539 


18709 


23317 


13630 


18790 


23390 


13722 


18871 


23462 


13813 


18952 


23535 


9.13904 


9.19033 


9.23607 


13994 


19113 


23679 


14085 


19193 


23752 


14175 


19273 


23823 


14266 


19353 


23895 


9.14356 


9.19433 


9.23967 


82° 


81° 

98° 


80° 


97° 


99° 



Sin 



60 
69 
58 
57 
56 
55 
54 
53 
53 
51 
50 
49 
48 
47 
46 
45 
44 
43 
43 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
23 
21 
20 
19 
18 
17 
16 
15 
14 
13 
13 
11 
10 
9 
8 
7 
6 
5 
4 
3 
3 
1 


Cos 



187 



li. LOGARITHMIC 





16y° 


168° 


167° 


166° 


165° 


bin 


10° 


11° 


12° 


13° 


14° 


0' 


9.23967 


9.28060 


9.31788 


9.35209 


9.38368 


1 


24039 


28125 


31847 


35263 


38418 


2 


24110 


28190 


31907 


35318 


38469 


3 


24181 


28254 


31966 


35373 


38519 


4 


24253 


28319 


32025 


35427 


38570 


6 


9.24324 


9.28384 


9.32084 


9.35481 


9.38620 


6 


24395 


28448 


32143 


35536 


38670 


7 


24466 


28512 


32202 


35590 


38721 


8 


24536 


28577 


32261 


35644 


38771 


9 


24607 


28641 


32319 


35698 


38821 


10 


9.24677 


9.28705 


9.32378 


9.35752 


9.38871 


11 


24748 


28769 


32437 


35806 


38921 


13 


24818 


28833 


32495 


35860 


38971 


13 


24888 


28896 


32553 


35914 


39021 


14 


24958 


28960 


32612 


35968 


39071 


16 


9.25028 


9.29024 


9.32670 


9.36022 


9.39121 


16 


25098 


29087 


32728 


36075 


39170 


17 


25168 


29150 


32786 


36129 


39220 


18 


25237 


29214 


32844 


36182 


39270 


19 


25307 


29277 


32902 


36236 


39319 


20 


9.25376 


9.29340 


9.32960 


9.36289 


9.39369 


21 


25445 


29403 


33018 


36342 


39418 


22 


25514 


29466 


33075 


36395 


39467 


23 


25583 


29529 


33133 


36449 


39517 


24 


25652 


29591 


33190 


36502 


39566 


25 


9.25721 


9.29654 


9.33248 


9.36555 


9.39615 


26 


25790 


29716 


33305 


36608 


39664 


27 


25858 


29779 


33362 


36660 


39713 


28 


25927 


29841 


33420 


36713 


39762 


29 


25995 


29903 


33477 


36766 


39811 


30 


9.26063 


9.29966 


9.33534 


9.36819 


9.39860 


31 


26131 


30028 


33591 


36871 


39909 


32 


26199 


30090 


33647 


36924 


39958 


33 


26267 


30151 


33704 


36976 


40006 


34 


26335 


30213 


33761 


37028 


40055 


35 


9.26403 


9.30275 


9.33818 


9.37081 


9.40103 


36 


26470 


30336 


33874 


37133 


40152 


37 


26538 


30398 


33931 


37185 


40200 


38 


26605 


30459 


33987 


37237 


40249 


39 


26672 


30521 


34043 


37289 


40297 


40 


9.26739 


9.30582 


9.34100 


9.37341 


9.40346 


41 


26806 


30643 


34156 


37393 


40394 


42 


26873 


30704 


34212 


37445 


40442 


43 


26940 


30765 


34268 


37497 


40490 


44 


27007 


30826 


34324 


37549 


40538 


45 


9.27073 


9.30887 


9.34380 


9.37600 


9.40586 


46 


27140 


30947 


34436 


37652 


40634 


47 


27206 


31008 


34491 


37703 


40682 


48 


27273 


31068 


34547 


37755 


40730 


49 


27339 


31129 


34602 


37806 


40778 


50 


9.27405 


9.31189 


9.34658 


9.37858 


9.40825 


51 


27471 


31250 


34713 


37909 


40873 


52 


27537 


31310 


34769 


37960 


40921 


53 


27602 


31370 


34824 


38011 


40968 


54 


27668 


31430 


34879 


38062 


41016 


65 


9.27734 


9.3I4dO 


9.34934 


9.38113 


9.41063 


56 


27799 


31549 


34989 


38164 


41111 


57 


27864 


31609 


35044 


38215 


41158 


58 


27930 


31669 


35099 


38266 


41205 


59 


27995 


31728 


35154 


38317 


41252 


60 


9.28060 


9.31788 


9.35209 


9.38368 


9.41300 




79^^ 


78° 


77° 


76° 


75° 


Cos 


100° 


101° 


102° 


103° 


104° 



188 



SINES AND COSINES 



164° 


163° 


162° 


161° 


160° 


Sin 


15° 


1«° 


17° 


18° 


iy° 




9.41300 


9.44034 


9.46594 


9.48998 


9.51264 


eo' 


41347 


44078 


46635 


49037 


51301 


59 


41394 


44122 


46676 


49076 


51338 


58 


41441 


44166 


46717 


49115 


51374 


57 


41488 


44210 


46758 


49153 


51411 


56 


9.41535 


9.44253 


9.46800 


9.49192 


9.51447 


55 


41582 


44297 


46841 


49231 


51484 


54 


41628 


44341 


46882 


49269 


51520 


53 


41675 


44385 


46923 


49308 


51557 


53 


41722 


44428 


46964 


49347 


51593 


51 


9.41768 


9.44472 


9.47005 


9.49385 


9.51629 


50 


41815 


44516 


47045 


49424 


51666 


49 


41861 


44559 


47086 


49462 


51702 


48 


41908 


44602 


47127 


49500 


51738 


47 


41954 


44646 


47168 


49539 


51774 


46 


9.42001 


9.44689 


9.47209 


9.49577 


9.51811 


45 


42047 


44733 


47249 


49615 


51847 


44 


42093 


44776 


47290 


49654 


51883 


43 


42140 


44819 


47330 


49692 


51919 


43 


42186 


44862 


47371 


49730 


51955 


41 


9.42232 


9.44905 


9.47411 


9.49768 


9.51991 


40 


42278 


44948 


47452 


49806 


52027 


39 


42324 


44992 


47492 


49844 


52063 


38 


42370 


45035 


47533 


49882 


52099 


37 


42416 


45077 


47573 


49920 


52135 


36 


9.42461 


9,45120 


9.47613 


9.49958 


9.52171 


35 


42507 


45163 


47654 


49996 


52207 


34 


42553 


45206 


47694 


50034 


52242 


33 


42599 


45249 


47734 


50072 


52278 


32 


42644 


45292 


47774 


50110 


52314 


31 


9.42690 


9.45334 


9.47814 


9.50148 


9.52350 


30 


42735 


45377 


47854 


50185 


52385 


29 


42781 


45419 


47894 


50223 


52421 


28 


42826 


45462 


47934 


50261 


52456 


27 


42872 


45504 


47974 


50298 


52492 


26 


9.42917 


9.45547 


9.48014 


9.50336 


9.52527 


25 


42962 


45589 


48054 


50374 


52563 


24 


43008 


45632 


48094 


50411 


52598 


23 


43053 


45674 


48133 


50449 


52634 


22 


43098 


45716 


48173 


50486 


52669 


21 


9.43143 


9.45758 


9.48213 


9.50523 


9.52705 


20 


43188 


45801 


48252 


50561 


52740 


19 


43233 


45843 


48292 


50598 


52775 


18 


43278 


45885 


48332 


50635 


52811 


17 


43323 


45927 


48371 


50673 


52846 


16 


9.43367 


9.45969 


9.48411 


9.50710 


9.52881 


15 


43412 


46011 


48450 


50747 


52916 


14 


43457 


46053 


48490 


50784 


52951 


13 


43502 


46095 


48529 


50821 


52986 


12 


43546 


46136 


48568 


50858 


53021 


11 


9.43591 


9.46178 


9.48607 


9.50896 


9.53056 


10 


43635 


46220 


48647 


50933 


53092 


9 


43680 


46262 


48686 


50970 


53126 


8 


43724 


46303 


48725 


51007 


53161 


7 


43769 


46345 


48764 


51043 


53196 


6 


9.43813 


9.46386 


9.48803 


9.51080 


9.53231 


5 


43857 


46428 


48842 


51117 


53266 


4 


43901 


46469 


48881 


51154 


53301 


3 


43946 


46511 


48920 


51191 


53336 


3 


43990 


46552 


48959 


51227 


53370 


1 


9.44034 


9.46594 


9.48998 


9.51264 


9.53405 





74° 


73° 


72° 


71° 


70° 


Cos 


105° 


106° 


107° 


108° 


109° 





189 



He LOGARITHMIC 



1 159° 


158° 


157° 


156° 


155° 


Sin 


20° 


31° 


33° 


33° 


34" 


C 


9.53405 


9.55433 


9.57358 


9.59188 


9.60931 


1 


53440 


55466 


57389 


59218 


60960 


9 


53475 


55499 


57420 


59247 


60988 


3 


53509 


55532 


57451 


59277 


61016 


4 


53544 


55564 


57482 


59307 


61045 


5 


9.53578 


9.55597 


9.57514 


9.59336 


9.61073 


6 


53613 


55630 


57545 


59366 


61101 


7 


53647 


55663 


57576 


59396 


61129 


8 


53682 


55695 


57607 


59425 


61158 


9 


53716 


55728 


57638 


59455 


61186 


10 


9.53751 


9.55761 


9.57669 


9.59484 


9.61214 


11 


53785 


55793 


57700 


59514 


61242 


12 


53819 


55826 


57731 


59543 


61270 


13 


53854 


55858 


57762 


59573 


61298 


14 


53888 


55891 


57793 


59602 


61326 


15 


9.53922 


9.55923 


9.57824 


9.59632 


9.61354 


16 


53957 


55956 


57855 


59661 


61382 


17 


53991 


55988 


57885 


59690 


61411 


18 


54025 


56021 


57916 


59720 


61438 


19 


54059 


56053 


57947 


59749 


61466 


20 


9.54093 


9.56085 


9.57978 


9.59778 


9.61494 


21 


54127 


56118 


58008 


59808 


61522 


23 


54161 


56150 


58039 


59837 


61550 


23 


54195 


56182 


5S070 


59866 


61578 


24 


54229 


56215 


58101 


59895 


61606 


25 


9.54263 


9.56247 


9.58131 


9.59924 


9.61634 


26 


54297 


56279 


58162 


59954 


61662 


27 


54331 


56311 


58192 


59983 


61689 


28 


54365 


56343 


58223 


60012 


61717 


29 


54399 


56375 


58253 


60041 


61745 


30 


9.54433 


9.56408 


9.58284 


9.60070 


9.61773 


31 


54466 


56440 


58314 


60099 


61800 


32 


54500 


56472 


58345 


60128 


61828 


33 


54534 


56504 


58375 


60157 


61856 


34 


54567 


56536 


58406 


60186 


61883 


35 


9.54601 


9.56568 


9.58436 


9.60215 


9.61911 


36 


54635 


56599 


58467 


60244 


61939 


37 


54668 


56631 


58497 


60273 


61966 


38 


54702 


56663 


58527 


60302 


61994 


39 


54735 


56695 


58557 


60331 


62021 


40 


9.54769 


9.56727 


9.58588 


9.60359 


9.62049 


41 


54802 


56759 


58618 


60388 


62076 


43 


54836 


56790 


58648 


60417 


62104 


43 


54869 


56822 


58678 


60446 


62131 


44 


54903 


56854 


58709 


60474 


62159 


45 


9.54936 


9.56886 


9.58739 


9.60503 


9.62186 


46 


54969 


56917 


58769 


60532 


62214 


47 


55003 


56949 


58799 


60561 


62241 


48 


55036 


56980 


58829 


60589 


62268 


49 


55069 


57012 


58859 


60618 


62296 


60 


9.55102 


9.57044 


9.58889 


9.60646 


9.62323 


51 


55136 


57075 


58919 


60675 


62350 


53 


55169 


57107 


58949 


60704 


62377 


53 


55202 


57138 


58979 


60732 


62405 


54 


55235 


57169 


59009 


60761 


62432 


55 


9.55268 


9.57201 


9.59039 


9.60789 


9.62459 


56 


55301 


57232 


59069 


60818 


62486 


57 


55334 


57264 


59098 


60846 


62513 


58 


55367 


57295 


59128 


00875 


62541 


59 


55400 


57326 


59158 


00903 


625G8 


60 


9.55433 


9.57358 


9.59188 


9.00931 


9.025C5 




69° 


68° 


67° 


66° 


65° 


Cos 


110° 


111° 


112° 


113° 


114° 



190 



SINES AND COSINES 



154° 


153° 


152° 


151° 


150° 


Sin 


25° 


26° 


27° 


28° 


29° 




9.62595 


9.64184 


9.65705 


9.67161 


9.68557 


6<y 


62622 


64210 


65729 


67185 


68580 


59 


62649 


64236 


65754 


67208 


68603 


58 


62676 


64262 


65779 


67232 


68625 


57 


62703 


64288 


65804 


67256 


68648 


56 


9.62730 


9.64313 


9.65828 


9.67280 


9.68671 


55 


62757 


64339 


65853 


67303 


68694 


54 


62784 


64365 


65878 


67327 


68716 


53 


62811 


64391 


65902 


67350 


68739 


52 


62838 


64417 


65927 


67374 


68762 


51 


9.62865 


9.64442 


9.65952 


9.67398 


9.68784 


50 


62892 


64468 


65976 


67421 


68807 


49 


• 62918 


64494 


66001 


67445 


68829 


48 


62945 


64519 


66025 


67468 


68852 


47 


62972 


64545 


66050 


67492 


68875 


46 


9.62999 


9.64571 


9.66075 


9.67515 


9.68897 


45 


63026 


64596 


66099 


67539 


68920 


44 


63052 


64622 


66124 


67562 


68942 


43 


63079 


64647 


66148 


67586 


68965 


43 


63106 


64673 


66173 


67609 


68987 


41 


9.63133 


9.64698 


9.66197 


9.67633 


9.69010 


40 


63159 


64724 


66221 


67656 


69032 


39 


63186 


64749 


66246 


67680 


69055 


38 


63213 


64775 


66270 


67703 


69077 


37 


63239 


64800 


66295 


67726 


69100 


36 


9.63266 


9.64826 


9.66319 


9.67750 


9.69122 


35 


63292 


64851 


66343 


67773 


69144 


34 


63319 


64877 


66368 


67796 


69167 


33 


63345 


64902 


66392 


67820 


69189 


32 


63372 


64927 


66416 


67843 


69212 


31 


9.63398 


9.64953 


9.66441 


9.67866 


9.69234 


30 


63425 


64978 


66465 


67890 


69256 


29 


63451 


65003 


66489 


67913 


69279 


28 


63478 


65029 


66513 


67936 


69301 


27 


63504 


65054 


66537 


67959 


69323 


26 


9.63531 


9.65079 


9.66562 


9.67982 


9.69345 


25 


63557 


65104 


66586 


68006 


69368 


24 


63583 


65130 


66610 


68029 


69390 


23 


63610 


65155 


66634 


68052 


69412 


22 


63636 


65180 


66658 


68075 


69434 


21 


9.63662 


9.65205 


9.66682 


9.68098 


9.69456 


20 


63689 


65230 


66706 


68121 


69479 


19 


63715 


65255 


66731 


68144 


69501 


18 


63741 


65281 


66755 


68167 


69523 


17 


63767 


65306 


66779 


68190 


69545 


16 


9.63794 


9.65331 


9.66803 


9.68213 


9.69567 


15 


63820 


65356 


66827 


68237 


69589 


14 


63846 


65381 


66851 


68260 


69611 


13 


63872 


65406 




68283 


69633 


12 


63898 


65431 


66899 


68305 


69655 


11 


9.63924 


9.65456 


9.66922 


9.68328 


9.69677 


10 


63950 


65481 


66946 


68351 


69699 


9 


63976 


65506 


66970 


68374 


69721 


8 


64002 


65531 


66994 


68397 


69743 


7 


64028 


65556 


67018 


68420 


69765 


6 


9.64054 


9.65580 


9.67042 


9.68443 


9.69787 


5 


64080 


65605 


67066 


68466 


69809 


4 


64106 


65630 


67090 


68489 


69831 


3 


64132 


65655 


67113 


68512 


69853 


2 


64158 


65680 


67137 


68534 


69875 


1 


9.64184 


9.65705 


9.67161 


9.68557 


9.69897 





64° 


63° 


62° 


61° 


60° 


Cos 


115° 


116° 


117° 


118° 


110° 





191 



II. LOGARITHMIC 





149° 


148° 


147° 


146° 


145° 


Sin 


30° 


31° 


33° 


33° 


34° 


(/ 


9.69897 


9.71184 


9.72421 


9.73611 


9.74756 


1 


69919 


71205 


72441 


73630 


74775 


2 


69941 


71226 


72461 


73650 


74794 


3 


69963 


71247 


72482 


73669 


74812 


4 


69984 


71268 


72502 


73689 


74831 


5 


9.70006 


9.71289 


9.72522 


9.73708 


9.74850 


6 


70028 


71310 


72542 


73727 


74868 


7 


70050 


71331 


72562 


73747 


74887 


8 


70072 


71352 


72582 


73766 


74906 


9 


70093 


71373 


72602 


73785 


74924 


10 


9.70115 


9.71393 


9.72622 


9.73805 


9.74943 


11 


70137 


71414 


72643 


73824 


74961 


13 


70159 


71435 


72663 


73843 


74980 


13 


70180 


71456 


72683 


73863 


74999 


14 


70202 


71477 


72703 


73882 


75017 


15 


9.70224 


9.71498 


9.72723 


9.73901 


9.75036 


16 


70245 


71519 


72743 


73921 


75054 


17 


70267 


71539 


72763 


73940 


75073 


18 


70288 


71560 


72783 


73959 


75091 


19 


70310 


71581 


72803 


73978 


75110 


30 


9.70332 


9.71602 


9.72823 


9.73997 


9.75128 


21 


70353 


71622 


72843 


74017 


75147 


23 


70375 


71643 


72863 


74036 


75165 


23 


70396 


71664 


72883 


74055 


75184 


24 


70418 


71685 


72902 


74074 


75202 


25 


9.70439 


9.71705 


9.72922 


9.74093 


9.75221 


26 


70461 


71726 


72942 


74113 


75239 


B 


70482 


71747 


72962 


74132 


75258 


^8 


70504 


71767 


72982 


74151 


75276 


29 


70525 


71788 


73002 


74170 


75294 


30 


9.70547 


9.71809 


9.73022 


9.74189 


9.75313 


31 


70568 


71829 


73041 


74208 


75331 


32 


70590 


71850 


73061 


74227 


75350 


33 


70611 


71870 


73081 


74246 


75368 


34 


70633 


71891 


73101 


74265 


75386 


35 


9.70654 


9.71911 


9.73121 


9.74284 


9.75405 


36 


70675 


71932 


73140 


74303 


75423 


37 


70697 


71952 


73160 


74322 


75441 


38 


70718 


71973 


73180 


74341 


75459 


39 


70739 


71994 


73200 


74360 


75478 


40 


9.70761 


9.72014 


9.73219 


9.74379 


9.75496 


41 


70782 


72034 


73239 


74398 


75514 


43 


70803 


72055 


73259 


74417 


75533 


43 


70824 


72075 


73278 


74436 


75551 


44 


70846 


72096 


73298 


74455 


75569 


45 


9.70867 


9.72116 


9.73318 


9.74474 


9.75587 


46 


70888 


72137 


73337 


74493 


75605 


47 


70909 


72157 


73357 


74512 


75624 


48 


70931 


72177 


73377 


74531 


75642 


49 


70952 


72198 


73396 


74649 


75660 


50 


9.70973 


9.72218 


9.73416 


9.74568 


9.75678 


51 


70994 


72238 


73435 


74587 


75696 


53 


71015 


72259 


73455 


74606 


75714 


53 


71036 


72279 


73474 


74625 


75733 


54 


71058 


72299 


73494 


74644 


75751 


55 


9.71079 


9.72320 


9.73513 


9.74662 


9.75769 


5§ 


71100 


72340 


73533 


74681 


75787 


57 


71121 


72360 


73552 


74700 


75805 


68 


71142 


72381 


73572 


74719 


75823 


59 


71163 


72401 


73591 


74737 


75841 


60 


9.71184 


9.72421 


9.73611 


9.74756 


9.75859 




59° 


58° 


57° 


56° 


55° 


Cos 


120° 


121° 


122° 


123° 


124° 



192 



sums AND COSINES 



144° 


143° 


142° 


141° 


140° 


Sin 


35° 


36° 


37° 


38° 


39° 




9.75859 


9.76922 


9.77946 


9.78934 


9.79887 


6(y 


75877 


76939 


77963 


78950 


79903 


69 


75895 


76957 


77980 


78967 


79918 


58 


75913 


76974 


77997 


78983 


79934 


57 


75931 


76991 


78013 


78999 


79950 


56 


9.75949 


9.77009 


9.78030 


9.79015 


9.79965 


55 


75967 


77026 


78047 


79031 


79981 


54 


75985 


77043 


78063 


79047 


79996 


53 


76003 


77061 


78080 


79063 


80012 


53 


76021 


77078 


78097 


79079 


80027 


51 


9.76039 


9.77095 


9.78113 


9.79095 


9.80043 


50 


76057 


77112 


78130 


79111 


80058 


49 


76075 


77130 


78147 


79128 


80074 


48 


76093 


77147 


78163 


79144 


80089 


47 


76111 


77164 


78180 


79160 


80105 


46 


9.76129 


9.77181 


9.78197 


9.79176 


9.80120 


45 


76146 


77199 


78213 


79192 


80136 


44 


76164 


• 77216 


78230 


79208 


80151 


43 


76182 


77233 


78246 


79224 


80166 


43 


76200 


77250 


78263 


79240 


80182 


41 


9.76218 


9.77268 


9.78280 


9.79256 


9.80197 


40 


76236 


77285 


78296 


79272 


80213 


39 


76253 


77302 


78313 


79288 


80228 


38 


76271 


77319 


78329 


79304 


80244 


37 


76289 


77336 


78346 


79319 


80259 


36 


9.76307 


9.77353 


9.78362 


9.79335 


9.80274 


35 


76324 


77370 


78379 


79351 


80290 


34 


76342 


77387 


78395 


79367 


80305 


33 


76360 


77405 


78412 


79383 


80320 


33 


76378 


77422 


78428 


79399 


80336 


31 


9.76395 


9.77439 


9.78445 


9.79415 


9.80351 


30 


76413 


77456 


78461 


79431 


80366 


39 


76431 


77473 


78478 


79447 


80382 


38 


76448 


77490 


78494 


79463 


80397 


37 


76466 


77507 


78510 


79478 


80412 


36 


9.76484 


9.77524 


9.78527 


9.79494 


9.80428 


35 


76501 


77541 


78543 


79510 


80443 


34 


76519 


77558 


78560 


79526 


80458 


33 


76537 


77575 


78576 


79542 


80473 


33 


76554 


77592 


78592 


79558 


80489 


31 


9.76572 


9.77609 


9.78609 


9.79573 


9.80504 


30 


76590 


77626 


78625 


79589 


80519 


19 


76607 


77643 


78642 


79605 


80534 


18 


76625 


77660 


78658 


79621 


80550 


17 


76642 


77677 


78674 


79636 


80565 


16 


9.76660 


9.77694 


9.78691 


9.79652 


9.80580 


15 


76677 


77711 


78707 


79668 


80595 


14 


76695 


77728 


78723 


79684 


80610 


13 


76712 


77744 


78739 


79699 


80625 


13 


76730 


77761 


78756 


79715 


80641 


11 


9.76747 


9.77778 


9.78772 


9.79731 


9.80656 


10 


76765 


77795 


78788 


79746 


80671 


9 


76782 


77812 


78805 


79762 


80686 


8 


76800 


. 77829 


78821 


79778 


80701 


7 


76817 


77846 


78837 


79793 


80716 


6 


9.76835 


9.77862 


9.78853 


9.79809 


9.80731 


5 


76852 


77879 


78869 


79825 


80746 


4 


76870 


77896 


78886 


79840 


80762 


3 


76887 


77913 


78902 


79856 


80777 


3 


76904 


77930 


78918 


79872 


80792 


1 


9.76922 


9.77946 


.78934 


9.79887 


9.80807 





54° 


53° 


53° 


51° 


50° 


Cos 


125° 


126° 


127° 


128° 


129° 





II. LOGARITHMIC 





139° 


138° 


137° 


136° 


135° 
44° 


Sin 


40° 


41° 


42° 


43° 


0' 


9.80807 


9.81694 


9.82551 


9.83378 


9.84177 


1 


80822 


81709 


82565 


83392 


84190 


2 


80837 


81723 


82579 


83405 


84203 


3 


80852 


81738 


82593 


83419 


84216 


4 


80867 


81752 


82607 


83432 


84229 


5 


9.80882 


9.81767 


9.82621 


9.83446 


9.84242 


6 


80897 


81781 


82635 


83459 


84255 


7 


80912 


81796 


82649 


83473 


84269 


8 


80927 


81810 


82663 


83486 


84282 


9 


80942 


81825 


82677 


83500 


84295 


10 


9.80957 


9.81839 


9.82691 


9.83513 


9.84308 


11 


80972 


81854 


82705 


83527 


84321 


12 


80987 


81868 


82719 


83540 


84334 


13 


81002 


81882 


82733 


83554 


84347 


14 


81017 


81897. 


82747 


83567 


84360 


15 


9.81032 


9.81911 


9.82761 


9.83581 


9.84373 


16 


81047 


81926 


82775 


83594 


84385 


17 


81061 


81940 


82788 


83608 


84398 


18 


81076 


81955 


82802 


83621 


84411 


19 


81091 


81969 


82816 


83634 


84424 


20 


9.81106 


9.81983 


9.82830 


9.83648 


9.84437 


21 


81121 


81998 


82844 


83661 


84450 


22 


81136 


82012 


82858 


83674 


84463 


23 


81151 


82026 


82872 


83688 


84476 


24 


81166 


82041 


82885 


83701 


84489 


25 


9.81180 


9.82055 


9.82899 


9.83715 


9.84502 


26 


81195 


82069 


82913 


83728 


84515 


27 


81210 


82084 


82927 


83741 


84528 


28 


81225 


82098 


82941 


83755 


84540 


29 


81240 


82112 


82955 


83768 


84553 


30 


9. 81254 


9.82126 


9.82968 


9.83781 


9.84566 


31 


81269 


82141 


82982 


83795 


84579 


32 


81284 


82155 


82996 


83808 


84592 


33 


81299 


82169 


83010 


83821 


84605 


34 


81314 


82184 


83023 


83834 


84618 


35 


9.81328 


9.82198 


9.83037 


9.83848 


9.84630 


36 


81343 


82212 


83051 


83861 


84643 


37 


81358 


82226 


83065 


83874 


84656 


38 


81372 


82240 


83078 


83887 


84669 


39 


81387 


82255 


83092 


83901 


84682 


40 


9.81402 


9.82269 


9.83106 


9.83914 


9.84694 


41 


81417 


82283 


83120 


83927 


84707 


42 


81431 


82297 


83133 


83940 


84720 


43 


81446 


82311 


83147 


83954 


84733 


44 


81461 


82326 


83161 


83967 


84745 


45 


9.81475 


9.82340 


9.83174 


9.83980 


9.84758 


46 


81490 


82354 


83188 


83993 


84771 


47 


81505 


82368 


83202 


84006 


84784 


48 


81519 


82382 


83215 


84020 


84796 


49 


81534 


82396 


83229 


84033 


84809 


50 


9.81549 


9.82410 


9.83242 


9.84046 


9.84822 


51 


81563 


82424 


83256 


84059 


84835 


52 


81578 


82439 


83270 


84072 


84847 


53 


81592 


82453 


83283 


84085 


84860 


54 


81607 


82467 


83297 


84098 


84873 


55 


9.81622 


9.82481 


9.83310 


9.84112 


9.84885 


56 


81636 


82495 


. 83324 


84125 


84898 


57 


81651 


82509 


83338 


84138 


84911 


58 


81665 


82523 


83351 


84151 


84923 


59 


81680 


82537 


83365 


84164 


84936 


60 


9.81694 


9.82551 


9.83378 


9.84177 


9.84949 




49° 


48° 


47° 


46° 


45° 


Cos 


130° 


131° 


132° 


133° 


134° 



194 



SINES AND COSINES 



134° 


133° 


132° 


131° 


130° 


Sin 


45° 


46° 


47° 


48° 


49° 




y. 84949 


9.85693 


9.86413 


9.87107 


9.87778 


6(y 


84961 


85706 


86425 


87119 


• 87789 


59 


84974 


85718 


86436 


87130 


87800 


58 


84986 


85730 


86448 


87141 


87811 


57 


84999 


85742 


86460 


87153 


87822 


56 


9.85012 


9.85754 


9.86472 


9.87164 


9.87833 


55 


85024 


85766 


86483 


87175 


87844 


54 


85037 


85779 


86495 


87187 


87855 


53 


85049 


85791 


86507 


87198 


87866 


52 


85062 


85803 


86518 


87209 


87877 


51 


9.85074 


9.85815 


9.86530 


9.87221 


9.87887 


50 


85087 


85827 


86542 


87232 


87898 


49 


85100 


85839 


86554 


87243 


87909 


48 


85112 


85851 


86565 


87255 


87920 


47 


85125 


85864 


86577 


87266 


87931 


46 


9.85137 


9.85876 


9.86589 


9.87277 


9.87942 


45 


85150 


85888 


86600 


87288 


87953 


44 


85162 


85900 


86612 


87300 


87964 


43 


85175 


85912 


86624 


87311 


87975 


43 


85187 


85924 


86635 


87322 


87985 


41 


9.85200 


9.85936 


9.86647 


9.873S4 


9.87996 


40 


85212 


85948 


86659 


87345 


88007 


39 


85225 


85960 


86670 


87356 


88018 


38 


85237 


85972 


86682 


87367 


88029 


37 


85250 


85984 


86694 


87378 


88040 


36 


9.85262 


9.85996 


9.86705 


9.87390 


9.88051 


35 


85274 


86008 


86717 


87401 


88061 


34 


85287 


86020 


86728 


87412 


88072 


33 


85299 


86032 


86740 


87423 


88083 


33 


85312 


86044 


86752 


87434 


88094 


31 


9.85324 


9.86056 


9.86763 


9.87446 


9.88105 


30 


85337 


86068 


86775 


87457 


88115 


29 


85349 


86080 


86786 


87468 


88126 


28 


85361 


86092 


86798 


87479 


88137 


37 


85374 


86104 


86809 


87490 


88148 


26 


9.85386 


9,86116 


9.86821 


9.87501 


9.88158 


25 


85399 


86128 


86832 


87513 


88169 


24 


85411 


86140 


86844 


87524 


881S0 


23 


85423 


86152 


86855 


87535 


88191 


22 


85436 


86164 


86867 


87546 


88201 


21 


9.85448 


9.86176 


9.86879 


9.87557 


9.88212 


20 


85460 


86188 


86890 


87568 


88223 


19 


85473 


86200 


86902 


87579 


88234 


18 


85485 


86211 


86913 


87590 


88244 


17 


85497 


86223 


86924 


87601 


88255 


16 


9.85510 


9.86235 


9.86936 


9.87613 


9.88266 


15 


85522 


86247 


86947 


87624 


88276 


14 


85534 


86259 


86959 


87635 


88287 


13 


85547 


86271 


86970 


87646 


88298 


12 


85559 


86283 


86982 


87657 


88308 


11 


9.85571 


9.86295 


9.86993 


9.87668 


9.88319 


10 


85583 


86306 


87005 


87679 


88330 


*> 


85596 


86318 


87016 


87690 


88340 


8 


85608 


86330 


87028 


87701 


88351 


7 


85620 


86342 


87039 


87712 


88362 


6 


9.85632 


9.86354 


9.87050 


9.87723 


9.88372 


5 


85645 


86366 


87062 


87734 


88383 


4 


85657 


86377 


87073 


87745 


8S394 


3 


85669 


86389 


87085 


87756 


88404 


2 


85681 


86401 


87096 


87767 


88415 


1 


9.85693 


9.86413 


9.87107 


9.87778 


9.88425 





44° 


43° 


42° 


41° 


40° 


Cos 


135° 


136° 


137° 


138° 


139° 





195 



11. LOGARITHMIC 



L 





129° 


128° 


127° 


126° 


125° 
54° 


Sin 


50° 


51° 


52° 


53° 


(y 


9.88425 


9.89050 


9.89653 


9.90235 


9.90796 


1 


88436 


89060 


89663 


90244 


90805 


2 


88447 


89071 


89673 


90254 


90814 


3 


88457 


89081 


89683 


90263 


90823 


4 


88468 


89091 


89693 


90273 


90832 


5 


9.88478 


9.89101 


9.89702 


9.90282 


9.90842 


6 


88489 


89112 


89712 


90292 


90851 


7 


88499 


89122 


89722 


90301 


90860 


8 


88510 


89132 


89732 


90311 


90869 


9 


88521 


89142 


89742 


90320 


90878 


10 


9.88531 


9.89152 


9.89752 


9.90330 


9.90887 


11 


88542 


89162 


89761 


90339 


90896 


13 


88552 


89173 


89771 


90349 


90906 


13 


88563 


89183 


89781 


90358 


90915 


14 


88573 


89193 


89791 


90368 


90924 


15 


9.88584 


9.89203 


9.89801 


9.90377 


9.90933 


16 


88594 


89213 


89810 


90386 


90942 


17 


88605 


89223 


89820 


90396 


90951 


18 


88615 


89233 


89830 


90405 


90960 


19 


88626 


89244 


89840 


90415 


90969 


20 


9. 88636 


9.89254 


9.89849 


9.90424 


9.90978 


21 


88647 


89264 


89859 


90434 


90987 


22 


88657 


89274 


89869 


90443 


90996 


23 


88668 


89284 


89879 


90452 


91005 


24 


88678 


89294 


89888 


90462 


91014 


25 


9.88688 


9.89304 


9.89898 


9.90471 


9.91023 


26 


88699 


89314 


89908 


90480 


91033 


27 


88709 


89324 


89918 


90490 


91042 


28 


88720 


89334 


89927 


90499 


91051 


29 


88730 


89344 


89937 


90509 


91060 


30 


9.88741 


9.89354 


9.89947 


9.90518 


9.91069 


31 


88751 


89364 


89956 


90527 


91078 


32 


88761 


89375 


89966 


90537 


91087 


33 


88772 


89385 


89976 


90546 


91096 


34 


88782 


89395 


89985 


90555 


91105 


35 


9.88793 


9.89405 


9.89995 


9.90565 


9.91114 


36 


88803 


89415 


90005 


90574 


91123 


37 


88813 


89425 


90014 


90583 


91132 


38 


88824 


89435 


90024 


90592 


91141 


39 


88834 


89445 


90034 


90602 


91149 


40 


9.88844 


9.89455 


9.90043 


9.90611 


9.91158 


41 


88855 


89465 


90053 


90620 


91167 


42 


88865 


89475 


90063 


90630 


91176 


43 


88875 


89485 


90072 


90639 


91185 


44 


88886 


89495 


90082 


90648 


91194 


45 


9.88896 


9.89504 


9.90091 


9.90657 


9.91203 


46 


88906 


89514 


90101 


90667 


91212 


47 


88917 


89524 


90111 


90676 


91221 


48 


88927 


89534 


90120 


90685 


91230 


49 


88937 


89544 


90130 


90694 


91239 


50 


9.88948 


9.89554 


9.90139 


9.90704 


9.91248 


51 


88958 


89564 


90149 


90713 


91257 


52 


88968 


89574 


90159 


90722 


91266 


53 


88978 


89584 


90168 


90731 


91274 


54 


88989 


89594 


90178 


90741 


91283 


55 


9.88999 


9.89604 


9.90187 


9.90750 


9.91292 


56 


89009 


89014 


90197 


90759 


91301 


57 


89020 


89624 


90206 


90768 


91310 


58 


89030 


89033 


90216 


90777 


91319 


59 


89040 


89643 


90225 


90787 


91328 


60 


9.89050 


9.89653 


9.90235 


9.90796 


9.91330 




39° 


38° 


37° 


36° 

143° 


35° 


Cos 1 


140° 


141° 


142° 


144° 



190 



SINES AND COSINES 



1 124° 


123° 


122° 


121° 


120° 


Sin 


55° 


56° 


57° 


58° 


59° 




9.91336 


9.91857 


9.92359 


9.92842 


9.93307 


60' 


91345 


91866 


92367 


92850 


93314 


59 


91354 


91874 


92376 


92858 


93322 


58 


91363 


91883 


92384 


92866 


93329 


57 


91372 


91891 


92392 


92874 


93337 


56 


9.91381 


9.91900 


9.92400 


9.92881 


9.93344 


55 


91389 


91908 


92408 


92889 


93352 


54 


91398 


91917 


92416 


92897 


93360 


53 


91407 


91925 


92425 


92905 


93367 


52 


91416 


91934 


92433 


92913 


93375 


51 


9.91425 


9.91942 


9.92441 


9. 92921 


9. 93382 


50 


91433 


91951 


92449 


92929 


93390 


49 


91442 


91959 


92457 


92936 


93397 


48 


91451 


91968 


92465 


92944 


93405 


47 


91460 


91976 


92473 


92952 


93412 


46 


9.91469 


9. 91985 


9. 92482 


9.92960 


9. 93420 


45 


91477 


91993 


92490 


92968 


93427 


44 


91486 


92002 


92498 


92976 


93435 


43 


91495 


92010 


92506 


92983 


93442 


43 


91504 


92018 


92514 


92991 


93450 


41 


9.91512 


9.92027 


9.92522 


9.92999 


9.93457 


40 


91521 


92035 


92530 


93007 


93465 


39 


91530 


92044 


92538 


93014 


93472 


38 


91538 


92052 


92546 


93022 


93480 


37 


91547 


92060 


92555 


93030 


93487 


36 


9.91556 


9.92069 


9.92563 


9.93038 


9.93495 


35 


91565 


92077 


92571 


93046 


93502 


34 


91573 


92086 


92579 


93053 


93510 


33 


91582 


92094 


92587 


93061 


93517 


33 


91591 


92102 


92595 


93069 


93525 


31 


9.91599 


9.92111 


9.92603 


9. 93077 


9.93532 


30 


91608 


92119 


92611 


93084 


93539 


29 


91617 


92127 


92619 


93092 


93547 


28 


91625 


92136 


92627 


93100 


93554 


27 


91634 


92144 


92635 


93108 


93562 


26 


9.91643 


9.92152 


9.92643 


9.93115 


9. 93569 


25 


91651 


92161 


92651 


93123 


93577 


24 


91660 


92169 


92659 


93131 


93584 


23 


91669 


92177 


92667 


93138 


93591 


22 


91677 


92186 


92675 


93146 


93599 


21 


9.91686 


9.92194 


9.92683 


9. 93154 


9.93606 


20 


91695 


92202 


92691 


93161 


93614 


19 


91703 


92211 


92699 


93169 


93621 


18 


91712 


92219 


92707 


93177 


93628 


17 


91720 


92227 


92715 


93184 


93636 


16 


9.91729 


9.92235 


9.92723 


9.93192 


9.93643 


15 


91738 


92244 


92731 


93200 


93650 


14 


91746 


92252 


92739 


93207 


93658 


13 


91755 


92260 


92747 


93215 


93665 


12 


91763 


92269 


92755 


93223 


93673 


11 


9.91772 


9.92277 


9.92763 


9.93230 


9.93680 


10 


91781 


92285 


92771 


93238 


93687 


9 


91789 


92293 


92779 


93246 


93695 


8 


91798 


92302 


92787 


93253 


93702 


7 


91806 


92310 


92795 


93261 


93709 


6 


9.91815 


9. 92318 


9. 92803 


9.93269 


9.93717 


5 


91823 


92326 


92810 


93276 


93724 


4 


91832 


92335 


92818 


93284 


93731 


3 


91840 


92343 


92826 


93291 


93738 


3 


91849 


92351 


92834 


93299 


93746 


1 


9.91857 


9.92359 


9.92842 


9. 93307 


9. 93753 





34° 


33° 


32° 


31° 


30° 

149° 


Cos 


. 145° 


146° 


147° 


148° 





197 



n. LOGARITHMIC 





1 119° 


118° 


117° 


116° 


115° 


Sin 


60° 


61° 


63° 


63° 


64° 


0' 


9.93753 


9.94182 


9.94593 


9.94988 


9.95366 


1 


93760 


94189 


94600 


94995 


95372 


3 


93768 


94196 


94607 


95001 


95378 


3 


93775 


94203 


94614 


95007 


95384 


4 


937S2 


94210 


94620 


95014 


95391 


5 


9.93789 


9.94217 


9.94627 


9.95020 


9.95397 


6 


93797 


94224 


94634 


95027 


95403 


7 


93804 


94231 


94640 


95033 


95409 


8 


93811 


94238 


94647 


95039 


95415 


9 


93819 


94245 


94654 


95046 


95421 


10 


9.93826 


9.94252 


9.94660 


9.95052 


9.95427 


11 


93833 


94259 


94667 


95059 


95434 


13 


93840 


94266 


94674 


95065 


95440 


13 


93847 


94273 


94680 


95071 


95446 


14 


93855 


94279 


94687 


95078 


95452 


15 


9.93862 


9.94286 


9.94694 


9.95084 


9.95458 


16 


93869 


94293 


94700 


95090 


95464 


17 


93876 


94300 


94707 


95097 


95470 


18 


93884 


94307 


94714 


95103 


95476 


19 


93891 


94314 


94720 


95110 


95482 


20 


9.93898 


9.94321 


9.94727 


9.95116 


9.95488 


21 


93905 


94328 


94734 


95122 


95494 


23 


93912 


94335 


94740 


95129 


95500 


23 


93920 


94342 


94747 


95135 


95507 


24 


93927 


94349 


94753 


95141 


95513 


25 


9. 93934 


9.94355 


9.94760 


9.95148 


9,95519 


26 


93941 


94362 


94767 


95154 


95525 


27 


93948 


94369 


94773 


95160 


95531 


28 


93955 


94376 


94780 


95167 


95537 


29 


93963 


94383 


94786 


95173 


95543 


30 


9.93970 


9.94390 


9.94793 


9.95179 


9.95549 


31 


93977 


94397 


94799 


95185 


95555 


33 


93984 


94404 


94806 


95192 


95561 


33 


93991 


94410 


94813 


95198 


95567 


34 


93998 


94417 


94819 


95204 


95573 


35 


9.94005 


9, 94424 


9.94826 


9.95211 


9.95579 


36 


94012 


94431 


94832 


95217 


95585 


37 


94020 


94438 


94839 


95223 


95591 


38 


94027 


94445 


94845 


95229 


95597 


39 


94034 


94451 


94852 


95236 


95603 


40 


9.94041 


9.94458 


9.94858 


9.95242 


9.95609 


41 


94048 


94465 


94865 


95248 


95615 


43 


94055 


94472 


94871 


95254 


95621 


43 


94062 


94479 


94878 


95261 


95627 


44 


94069 


94485 


94885 


95267 


95633 


45 


9.94076 


9. 94492 


9.94891 


9.95273 


9.95639 


46 


94083 


94499 


94898 


95279 


95645 


47 


94090 


94506 


94904 


95286 


95651 


48 


94098 


94513 


94911 


95292 


95657 


49 


94105 


94519 


94917 


95298 


95663 


50 


9.94112 


9. 94526 


9.94923 


9.95304 


9.95668 


61 


94119 


94533 


94930 


95310 


95674 


63 


94126 


94540 


94936 


95317 


95680 


63 


94133 


94546 


94943 


95323 


95686 


54 


94140 


94553 


94949 


95329 


95692 


55 


9.94147 


9. 94560 


9.94956 


9.95335 


9.95698 


66 


94154 


94567 


94962 


95341 


95704 


67 


94161 


94573 


94969 


95348 


95710 


58 


94168 


94580 


94975 


95354 


95716 


59 


94175 


94587 


94982 


95360 


95722 


60 


9.94182 


9.94593 


9.94988 


9.95366 


9.95728 




39° 


38° 


37° 


36° 


36° 


Cos 


150° 


151° 


152° 


153° 


154° 



198 



SmES AND COSINES 



113° 


112° 


111° 


110° 
69° 


66° 


67° 


68° 


9.96073 


9.96403 


9.96717 


9.97015 


96079 


96408 


96722 


97020 


96084 


96413 


96727 


97025 


96090 


96419 


96732 


97030 


96095 


96424 


96737 


97035 


9.96101 


9.96429 


9.96742 


9.97039 


96107 


96435 


96747 


97044 


96112 


96440 


96752 


97049 


96118 


96445 


96757 


97054 


96123 


96451 


96762 


97059 


9.96129 


9.96456 


9.96767 


9.97063 


96135 


96461 


96772 


97068 


96140 


96467 


96778 


97073 


96146 


96472 


96783 


97078 


96151 


96477 


96788 


97083 


9.96157 


9.96483 


9.96793 


9.97087 


96162 


96488 


96798 


97092 


96168 


96493 


96803 


97097 


96174 


96498 


96808 


97102 


96179 


96504 


96813 


97107 


9.96185 


9. 96509 


9.96818 


9.97111 


96190 


96514 


96823 


97116 


96196 


96520 


96828 


97121 


96201 


96525 


96833 


97126 


96207 


96530 


96838 


97130 


9.96212 


9.96535 


9.96843 


9.97135 


96218 


96541 


96848 


97140 


96223 


96546 


96853 


97145 


96229 


96551 


96858 


97149 


96234 


96556 


96863 


97154 


9.96240 


9.96562 


9. 96868 


9.97159 


96245 


96567 


96873 


97163 


96251 


96572 


96878 


97168 


96256 


96577 


96883 


97173 


96262 


96582 


96888 


97178 


9.96267 


9.96588 


9.96893 


9.97182 


06273 


96593 


96898 


97187 


96278 


96598 


96903 


97192 


96284 


96603 


96907 


97196 


96289 


96608 


96912 


97201 


9.96294 


9.96614 


9.96917 


9.97206 


96300 


96619 


96922 


97210 


96305 


96624 


96927 


97215 


96311 


96629 


96932 


97220 


96316 


96634 


96937 


97224 


9.96322 


9.96640 


9.96942 


9.97229 


96327 


96645 


96947 


97234 


96333 


96650 


96952 


97238 


96338 


96655 


96957 


97243 


96343 


96660 


96962 


97248 


9.96349 


9.96665 


9.96966 


9.97252 


96354 


96670 


96971 


97257 


96360 


96676 


96976 


97262 


96365 


96681 


96981 


97266 


96370 


96686 


96986 


97271 


9.96376 


9.96691 


9.96991 


9. 97276 


96381 


. 96696 


96996 


97280 


96387 


96701 


97001 


97285 


96392 


96706 


97005 


97289 


96397 


96711 


97010 


97294 


9.96403 


9.96717 


9.97015 


9.97299 


23° 


32° 


21° 


20° 


156° 


157° 


158° 


159° 



Sin 



6(y 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
13 
11 
10 
9 
8 
7 
6 
5 
4 
3 
3 
1 


Cos 



199 



^' 



II. LOGARITHMIC 





109° 


108° 

71° 


107° 


106° 


105° 


Sin 


70° 


73° 


73° 


74° 


0' 


9.97299 


9.97567 


9.97821 


9.98060 


9.98284 


1 


97303 


97571 


97825 


98063 


98288 


3 


97308 


97576 


97829 


98067 


98291 


3 


97312 


97580 


97833 


98071 


98295 


4 


97317 


97584 


97837 


98075 


98299 


5 


9. 97322 


9.97589 


9. 97841 


9.98079 


9.98302 


6 


97326 


97593 


97845 


98083 


98306 


7 


97331 


97597 


97849 


98087 


98309 


8 


97335 


97602 


97853 


98090 


98313 


9 


97340 


97606 


97857 


98094 


98317 


10 


9. 97344 


9.97610 


9.97861 


9.98098 


9.98320 


11 


97349 


97615 


97866 


98102 


98324 


13 


97353 


97619 


97870 


98106 


98327 


13 


97358 


. 97623 


97874 


98110 


98331 


14 


97363 


97628 


97878 


98113 


98334 


15 


9.97367 


9.97632 


9. 97882 


9.98117 


9.98338 


16 


97372 


97636 


97886 


98121 


98342 


17 


97376 


97640 


97890 


98125 


98345 


18 


97381 


97645 


97894 


98129 


98349 ; 


19 


97385 


97649 


97898 


98132 


98352 ; 


20 


9.97390 


9.97653 


9. 97902 


9.98136 


9.98356 


31 


97394 


97657 


97906 


98140 


98359 ' 


33 


97399 


97662 


97910 


98144 


98363 ; 


33 


97403 


97666 


97914 


98147 


98366 


34 


97408 


97670 


97918 


98151 


98370 \ 


35 


9.97412 


9.97674 


9.97922 


9.98155 


9.98373 ; 


36 


97417 


97679 


97926 


98159 


98377 


37 


97421 


97683 


97930 


98162 


98381 ! 


38 


97426 


97687 


97934 


98166 


98384 


39 


97430 


97691 


97938 


98170 


98388 1 


30 


9.97435 


9.97696 


9.97942 


9.98174 


9.98391 


31 


97439 


97700 


97946 


98177 


98395 


33 


97444 


97704 


97950 


98181 


98398 i 


33 


97448 


97708 


97954 


98185 


98402 ! 


34 


97453 


97713 


97958 


98189 


98405 i 


35 


9.97457 


9.97717 


9.97962 


9.98192 


9.98409 


36 


97461 


97721 


97966 


98196 


98412 


37 


97466 


97725 


97970 


98200 


98415 


38 


97470 


97729 


97974 


98204 


98419 


39 


97475 


97734 


97978 


98207 


98422 


40 


9. 97479 


9.97738 


9. 97982 


9.98211 


9.98426 


41 


97484 


97742 


97986 


98215 


98429 


43 


97488 


97746 


97989 


98218 


98433 


43 


97492 


97750 


97993 


98222 


98436 


44 


97497 


97754 


97997 


98226 


98440 ; 


45 


9.97501 


9.97759 


9.98001 


9.98229 


9.98443 i 


46 


97506 


97763 


98005 


98233 


98447 \ 


47 


97510 


97767 


98009 


98237 


98450 


48 


97515 


97771 


98013 


98240 


98453 : 


49 


97519 


97775 


98017 


98244 


98457 j 


50 


9. 97523 


9.97779 


9.98021 


9.98248 


9.98460 f 


51 


97528 


97784 


98025 


98251 


98464 
98467 1 


53 


97532 


97788 


98029 


98255 


53 


97536 


97792 


98032 


98259 


98471 : 


54 


97541 


97796 


98036 


98262 


98474 


5^ 


9. 97545 


9.97800 


9.98040 


9. 98266 


9. 98477 1 


56 


97550 


97804 


98044 


98270 


98481 


57 


97554 


97808 


98048 


98273 


98484 


58 


97558 


97812 


98052 


98277 


98488 1 


59 


97563 


97817 


98056 


98281 


98491 1 


60 


9.97567 


9.97821 


9.98060 


9. 982S4 


9.98494 




19° 

160° 


18° 


17° 


16° 


15° 


Cos 


161° 


162° 


163° 


164° 


200 i 

1 



SINES AND COSINES 



104° 


103° 


102° 


101° 


100° 


Sin 


75° 


76° 


77° 


78° 


79° 




9.98494 


9.98690 


9.98872 


9.99040 


9.99195 


ecy 


98498 


98694 


98875 


99043 


99197 


59 


98501 


98697 


98878 


99046 


99200 


58 


98505 


98700 


98881 


99048 


99202 


57 


98508 


98703 


98884 


99051 


99204 


56 


9.98511 


9.98706 


9.98887 


9.99054 


9.99207 


55 


98515 


98709 


98890 


99056 


99209 


54 


98518 


98712 


98893 


99059 


99212 


53 


98521 


98715 


98896 


99062 


99214 


52 


98525 


98719 


98898 


99064 


99217 


51 


9.98528 


9.98722 


9.98901 


9.99067 


9.99219 


50 


98531 


98725 


98904 


9907C 


99221 


49 


98535 


98728 


98907 


99072 


99224 


48 


98538 


98731 


98910 


99075 


99226 


47 


98541 


98734 


98913 


99078 


99229 


46 


9.98545 


9.98737 


9.98916 


9.99080 


9.99231 


45 


98548 


98740 


98919 


99083 


99233 


44 


98551 


98743 


98921 


99086 


99236 


43 


98555 


98746 


98924 


99088 


99238 


42 


98558 


98750 


98927 


99091 


99241 


41 


9.98561 


9.98753 


9.98930 


9.99093 


9.99243 


40 


98565 


98756 


98933 


99096 


99245 


39 


98568 


98759 . 


98936 


99099 


99248 


38 


98571 


98762 


98938 


99101 


99250 


37 


98574 


98765 


98941 


99104 


99252 


36 


9.98578 


9.98768 


9.98944 


9.99106 


9.99255 


35 


98581 


98771 


98947 


99109 


99257 


34 


98584 


98774 


98950 


99112 


99260 


33 


98588 


98777 


98953 


99114 


99262 


32 


98591 


98780 


98955 


99117 


99264 


31 


9. 98594 


9.98783 


9.98958 


9.99119 


9.99267 


30 


98597 


98786 


98961 


99122 


99269 


29 


98601 


98789 


98964 


99124 


99271 


28 


98604 


98792 


98967 


99127 


99274 


27 


98607 


98795 


98969 


99130 


99276 


26 


9.98610 


9.98798 


9.98972 


9.99132 


9.99278 


25 


98614 


98801 


98975 


99135 


99281 


24 


98617 


98804 


98978 


99137 


99283 


23 


98620 


98807 


98980 


99140 


99285 


22 


98623 


98810 


98983 


99142 


99288 


21 


9. 98627 


9.98813 


9.98986 


9.99145 


9.99290 


20 


98630 


98816 


98989 


99147 


99292 


19 


98633 


98819 


98991 


99150 


99294 


18 


98636 


98822 


98994 


99152 


99297 


17 


98640 


98825 


98997 


99155 


99299 


16 


9. 98643 


9.98828 


9.99000 


9.99157 


9.99301 


15 


98646 


98831 


99002 


99160 


99304 


14 


98649 


98834 


99005 


99162 


99306 


13 


98652 


98837 


99008 


99165 


99308 


12 


98656 


98840 


99011 


99167 


99310 


11 


9. 98659 


9. 98843 


9.99013 


9.99170 


9.99313 


10 


.98662 


98846 


99016 


99172 


99315 


9 


98665 


98849 


99019 


99175 


99317 


8 


98668 


98852 


99022 


99177 


99319 


7 


98671 


98855 


99024 


99180 


99322 


6 


9. 98675 


9.98858 


9.99027 


3.99182 


9.99324 


5 


98678 


98861 


99030 


99185 


99326 


4 


98681 


98864 


99032 


99187 


99328 


3 


98684 


98867 


99035 


99190 


99331 


2 


98687 


98869 


99038 


99192 


99333 


1 


9.98690 


9.98872 


9.99040 


9.99195 


9.99335 





14° 


13° 


12° 


11° 


10° 


Cos 


165° 


166° 


167° 


168° 


169° 





201 



II. LOGARITHMIC 





99° 


98° 


97° 


96° 


95° 


Sin 


80° 


81° 


83° 


83° 


84° 


0' 


9.99335 


9.99462 


9.99575 


9.99675 


9.99761 


1 


99337 


99464 


99577 


99677 


99763 


2 


99340 


99466 


99579 


99678 


99764 


3 


99342 


99468 


99581 


99680 


99765 


4 


99344 


99470 


99582 


99681 


99767 


5 


9.99346 


9.99472 


9.99584 


9.99683 


9.99768 


6 


99348 


99474 


99586 


99684 


99769 


7 


99351 


99476 


99588 


99686 


99771 


8 


99353 


99478 


99589 


99687 


99772 


9 


99355 


99480 


99591 


99689 


99773 


10 


9.99357 


9.99482 


9.99593 


9.99690 


9.99775 


11 


99359 


99484 


99595 


99692 


99776 


13 


99362 


99486 


99596 


99693 


99777 


13 


99364 


99488 


99598 


99695 


99778 


14 


99366 


99490 


99600 


99696 


99780 


15 


9.99368 


9. 99492 


9.99601 


9.99698 


9.99781 


16 


99370 


99494 


99603 


99699 


99782 


17 


99372 


99495 


99605 


99701 


99783 


18 


99375 


99497 


99607 


99702 


99785 


19 


99377 


99499 


99608 


99704 


99786 


30 


9.99379 


9.99501 


9.99610 


9.99705 


9.99787 


31 


99381 


99503 


99612 


99707 


99788 


33 


99383 


99505 


99613 


99708 


99790 


33 


99385 


99507 


99615 


99710 


99791 


34 


99388 


99509 


99617 


99711 


99792 


35 


9.99390 


9.99511 


9.99618 


9.99713 


9.99793 


36 


99392 


99513 


99620 


99714 


99795 


37 


99394 


99515 


99622 


99716 


99796 


38 


99396 


99517 


99624 


99717 


99797 


39 


99398 


99518 


99625 


99718 


99798 


30 


9.99400 


9. 99520 


9.99627 


9.99720 


9.99800 


31 


99402 


99522 


99629 


99721 


99801 


33 


99404 


99524 


99630 


99723 


99802 


. 33 


99407 


99526 


99632 


99724 


99803 


34 


99409 


99528 


99633 


99726 


99804 


35 


9.99411 


9.99530 


9.99635 


9.99727 


9.99806 


38 


99413 


99532 


99637 


99728 


99807 


37 


99415 


99533 


99638 


99730 


99808 


38 


99417 


99535 


99640 


99731 


99809 


39 


99419 


99537 


99642 


99733 


99810 


40 


9.99421 


9. 99539 


9.99643 


9.99734 


9.99812 


41 


99423 


99541 


99645 


99736 


99813 


43 


99425 


99543 


99647 


99737 


99814 


43 


99427 


99545 


99648 


99738 


99815 


44 


99429 


99546 


99650 


99740 


99816 


45 


9.99432 


9.99548 


9.99651 


9.99741 


9.99817 


46 


99434 


99550 


99653 


99742 


99819 


47 


99436 


99552 


99655 


99744 


99820 


48 


99438 


99554 


99656 


99745 


99821 


49 


99440 


99556 


99658 


99747 


99822 


50 


9.99442 


9.99557 


9.99659 


9.99748 


9.99823 


51 


99444 


99559 


99661 


99749 


99824 


53 


99446 


99561 


99663 


99751 


99825 


53 


99448 


99563 


99664 


99752 


99827 


54 


99450 


99565 


99666 


99753 


99828 


55 


9.99452 


9. 99566 


9.99667 


9.99755 


9.99829 


56 


99454 


99508 


99669 


99756 


99830 


57 


99456 


99570 


99670 


99757 


99831 


58 


99458 


99572 


99672 


99759 


99832 


59 


99460 


99574 


99674 


99760 


99833 


60 


9.99462 


9.99575 


9.99675 


9.99761 


0.99834 




9° 


8° 


7° 


6° 

173° 


5° 


Cos 


170° 


171° 


172° 


174° 



202 



SINES AND COSINES 



^^ 



94° 


93° 


92° 


91° 


90° 1 


Sin 


85° 


86° 


87° 


88= 


89° 




9.99834 


9.99894 


9.99940 


9. 99974 


9.99993 


ey 


99836 


99895 


99941 


99974 


99994 


5i) 


99837 


99896 


99942 


99974 


99994 


58 


99838 


99897 


99942 


99975 


99994 


57 


99839 


99898 


99943 


99975 


99994 


56 


9.99S40 


9.99898 


9.99944 


9.99976 


9.99994 


55 


99841 


99899 


99944 


99976 


99995 


54 


99842 


99900 


99945 


99977 


99995 


53 


99843 


99901 


99946 


99977 


99995 


53 


99844 


99902 


99946 


99977 


99995 


51 


9.99845 


9.99903 


9.99947 


9.99978 


9.99995 


50 


99846 


99904 


99948 


99978 


99996 


49 


99847 


99904 


99948 


99979 


99996 


48 


99848 


99905 


99949 


99979 


99996 


47 


99850 


99906 


99949 


99979 


99996 


46 


9.99851 


9.99907 


9.99950 


9. 99980 


9.99996 


45 


99852 


99908 


99951 


99980 


99996 


44 


99853 


99909 


99951 


99981 




43 


99854 


99909 


99952 


99981 


99997 


43 


99855 


99910 


99952 


99981 


99997 


41 


9.99856 


9.99911 


9.99953 


9.99982 


9.99997 


40 


99857 


99912 


99954 


99982 


99997 


39 


99858 


99913 


99954 


99982 


99997 


38 


99859 


99913 


99955 


99983 


99997 


37 


99860 


99914 


99955 


99983 


99998 


36 


9.99861 


9.99915 


9. 99956 


9.99983 


9.99998 


35 


99S62 


99916 


99956 


99984 


99998 


34 


99863 


99917 


99957 


99984 


99998 


33 


99864 


99917 


99958 


99984 


99998 


33 


99865 


99918 


99958 


99985 


99998 


31 


9.99866 


9.99919 


9. 99959 


9.99985 


9.99998 


30 


99867 


99920 


99959 


99985 


99998 


29 


99868 


99920 


99960 


99986 


99999 


28 


99869 


99921 


99960 


99986 


99999 


37 


99870 


99922 


99961 


99986 


99999 


26 


9.99871 


9. 99923 


9. 99961 


9.99987 


9.99999 


35 


99872 ■ 


99923 


99962 


99987 


99999 


24 


99873 


99924 


99962 


99987 


99999 


33 


99874 


99925 


99963 


99988 


99999 


33 


99875 


99926 


99963 


99988 


99999 


31 


9.99876 


9.99926 


9.99964 


9.99988 


9. 99999 


30 


99877 


99927 


99964 


99989 


99999 


19 


99878 


99928 


99965 


99989 


99999 


18 


99879 


99929 


99966 


99989 


99999 


17 


99879 


99929 


99966 


99989 


00000 


16 


9.99880 


9.99930 


9.99967 


9. 99990 


0. 00000 


15 


99881 


99931 


99967 


99990 


00000 


14 


99882 


99932 


99967 


99990 


00000 


13 


99883 


99932 


99968 


99990 


00000 


12 


99884 


99933 


99968 


99991 


00000 


11 


9.99885 


9. 99934 


9.99969 


9 99991 


0.00000 


10 


99886 


99934 


99969 


99991 


00000 


9 


99887 


99935 


99970 


99992 


00000 


8 


99888 


99936 


99970 


99992 


00000 


7 


998S9 


99936 


99971 


99992 


00000 


6 


9.99890 


9.99937 


9.99971 


9. 99992 


0. 00000 


5 


99891 


99938 


99972 


99992 


00000 


4 


99891 


99938 


99972 


99993 


00000 




99892 


99939 


99973 


99993 


00000 


2 


99893 


99940 


99973 


99993 


00000 


1 


9.99894 


9.99940 


9.99974 


9.99993 


0. 00000 





40 


3° 


2° 
177° 


1° 

178° 


0° 


Cos 


175° 


176° 


179° 





203 









III. 


LOGARITHMIC 




179° 


178° 


177° 


176° 


175° 


Tan 


0° 


1° 


3° 


3° 


4° 


0' 


— 00 


8.24192 


8.54308 


8.71940 


8.84464 


1 


6.46373 


24910 


54669 


72181 


84646 


2 


76476 


25616 


55027 


72420 


84826 


3 


94085 


26312 


55382 


72659 


85006 


4 


7.06579 


26996 


55734 


72896 


85185 


5 


7.16270 


8.27669 


8.56083 


8.73132 


8.85363 


6 


24188 


28332 


56429 


73366 


85540 


7 


30882 


28986 


56773 


73600 


85717 


8 


36682 


29629 


57114 


73S32 


85893 


9 


41797 


30263 


57452 


74063 


86069 


10 


7.46373 


8.30888 


8.57788 


8.74292 


8.86243 


11 


50512 


31505 


58121 


74521 


86417 


13 


54291 


32112 


58451 


74748 


86591 


13 


57767 


32711 


58779 


74974 


86763 


14 


60986 


33302 


59105 


75199 


86935 


15 


7.63982 


8.33886 


8.59428 


8.75423 


8.87106 


16 


66785 


34461 


59749 


75645 


87277 


17 


69418 


35029 


60068 


75867 


87447 


18 


71900 


35590 


603S4 


76087 


87616 


19 


74248 


36143 


60698 


76306 


87785 


30 


7.76476 


8.36689 


8-61009 


8.76525 


8.87953 


31 


78595 


37229 


61319 


76742 


88120 


33 


80615 


37762 


61626 


76958 


88287 


33 


82548 


38289 


61931 


77173 


88453 


34 


84394 


38809 


62234 


77387 


88618 


35 


7.86167 


8.39323 


8.62535 


8.77600 


8.88783 


36 


87871 


39832 


62834 


77811 


88948 


37 


89510 


40334 


63131 


78022 


89111 


38 


91089 


40830 


63426 


78232 


89274 


39 


92613 


41321 


63718 


78441 


89437 


30 


7.94086 


8.41807 


8.64009 


8.78649 


8.89598 


31 


95510 


42287 


64298 


78855 


89760 


33 


96889 


42762 


64585 


79061 


89920 


33 


98225 


43232 


64870 


79266 


90080 


34 


99522 


43696 


65154 


79470 


90240 


35 


8.00781 


8.44156 


8.65435 


8.79673 


8.90399 


36 


02004 


44611 


65715 


79875 


90557 


37 


03194 


45061 


65993 


80076 


90715 


38 


04353 


45507 


66269 


80277 


90872 


39 


05481 


45948 


66543 


80476 


91029 


40 


8.06581 


8.40385 


8.66816 


8.80674 


8.91185 


41 


07653 


46817 


67087 


80872 


91340 


43 


08700 


47245 


67356 


81068 


91495 


43 


09722 


47669 


67624 


81264 


91650 


44 


10720 


48089 


67890 


81459 


91803 


45 


8.11696 


8.48505 


8.68154 


8.81653 


8.91957 


46 


12651 


48917 


68417 


81846 


92110 


47 


13585 


49325 


68678 


82038 


92262 


48 


14500 


49729 


68938 


82230 


92414 


49 


15395 


50130 


69196 


82420 


92565 


50 


8.16273 


8.50527 


8.69453 


8.82610 


8.92716 


51 


17133 


50920 


69708 


82799 


92866 


63 


17976 


51310 


69962 


82987 


93016 


53 


18804 


51696 


70214 


83175 


93165 


54 


19616 


52079 


70465 


83361 


93313 


55 


8.20413 


8.52459 


8.70714 


8.83547 


8.93462 


56 


21195 


52835 


70962 


83732 


93609 


57 


21964 


53208 


71208 


83916 


93756 


58 


22720 


53578 


71453 


84100 


93903 


59 


23462 


53945 


71697 


84282 


94049 


60 


8.24192 


8.54308 


8.71940 


8.84464 


8.94195 




89° 


88° 


87° 


86° 


85° 


Cot 


90° 


91° 


92° 


93° 


94° 



204 



TANGENTS AND COTANGENTS 



^v^ 



174° 


173° 


172° 


171° 


170° 


Tan 


5° 


6° 


7° 


8° 


9° 




8.94195 


9.02162 


9.08914 


9.14780 


9.19971 


6(y 


94340 


02283 


09019 


14872 


20053 


59 


94485 


02404 


09123 


14963 


20134 


58 


94630 


02525 


09227 


15054 


20216 


57 


94773 


02645 


09330 


15145 


20297 


56 


8.94917 


9.02766 


9.09434 


9.15236 


9.20378 


55 


95060 


02885 


09537 


15327 


20459 


54 


95202 


03005 


09640 


15417 


20540 


53 


95344 


03124 


09742 


15508 


20621 


53 


95486 


03242 


09845 


15598 


20701 


51 


8. 95627 


9.03361 


9.09947 


9.15688 


9.20782 


50 


95767 


03479 


10049 


15777 


20862 


49 


95908 


03597 


10150 


15867 


20942 


48 


96047 


03714 


10252 


15956 


21022 


47 


96187 


03832 


10353 


16046 


21102 


46 


8.96325 


9.03948 


9.10454 


9.16135 


9.21182 


45 


96464 


04065 


10555 


16224 


21261 


44 


96602 


04181 


10656 


16312 


21341 


43 


96739 


04297 


10756 


16401 


21420 


43 


96877 


04413 


10856 


16489 


21499 


41 


8.97013 


9.04528 


9.10956 


9.16577 


9.21578 


40 


97150 


04643 


11056 


16665 


21657 


39 


97285 


04758 


11155 


16753 


21736 


38 


97421 


04873 


11254 


16841 


21814 


37 


97556 


04987 


11353 


16928 


21893 


36 


8.97691 


9.05101 


9.11452 


9.17016 


9.21971 


35 


97825 


05214 


11551 


17103 


22049 


34 


97959 


05328 


11649 


17190 


22127 


33 


98092 


05441 


11747 


17277 


22205 


33 


98225 


05553 


11845 


17363 


22283 


31 


8.98358 


9.05666 


9.11943 


9.17450 


9.22361 


30 


98490 


05778 


12040 


17536 


22438 


39 


98622 


05890 


12138 


17622 


22516 


38 


98753 


06002 


12235 


17708 


22593 


37 


98884 


06113 


12332 


17794 


22670 


36 


8.99015 


9.06224 


9.12428 


9.17880 


9.22747 


35 


99145 


06335 


12525 


17965 


22824 


34 


99275 


06445 


12621 


18051 


22901 


33 


99405 


06556 


12717 


18136 


22977 


33 


99534 


06666 


12813 


18221 


23054 


31 


8.99662 


9.06775 


9.12909 


9.18306 


9.23130 


30 


99791 


06885 


13004 


18391 


23206 


19 


99919 


06994 


13099 


18475 


23283 


18 


9.00046 


07103 


33194 


18560 


23359 


17 


00174 


07211 


13289 


18644 


23435 


16 


9.00301 


9.07320 


9.13384 


9.18728 


9.23510 


15 


00427 


07428 


13478 


18812 


23586 


14 


00553 


07536 


13573 


18896 


23661 


13 


00679 


07643 


13667 


18979 


23737 


13 


00805 


07751 


13761 


19063 


23812 


11 


9.00930 


9.07858 


9.13854 


9.19146 


9.23887 


10 


01055 


07964 


13948 


19229 


23962 


9 


01179 


08071 


14041 


19312 


24037 


8 


01303 


08177 


14134 


19395 


24112 


7 


01427 


08283 


14227 


19478 


24186 


6 


9.01550 


9.08389 


9.14320 


9.19561 


9.24261 


5 


01673 


08495 


14412 


19643 


24335 


4 


01796 


08600 


14504 


19725 


24410 


3 


01918 


08705 


14597 


19807 


24484 


3 


02040 


08810 


14688 


19S89 


24558 


1 


9.02162 


9.08914 
83° 


9.14780 


9.19971 


9.24632 





84° 


83° 


81° 


80° 


Cot 


95° 


96° 


97° 


98° 


99° 





205 



III. LOGARITHMIC 





169° 


168° 


167° 


166° 


165° 


Tan 


10° 


11° 


13° 


13° 


14° 


0' 


9.24632 


9.28865 


9.32747 


9.36336 


9.39677 


1 


24706 


28933 


32810 


36394 


39731 


2 


24779 


29000 


32872 


36452 


39785 


3 


24853 


29067 


32933 


36509 


39838 


4 


24926 


29134 


32995 


36566 


39892 


5 


9.25000 


9.29201 


9.33057 


9.36624 


9.39945 


6 


25073 


29268 


33119 


36681 


39999 


7 


25146 


29335 


33180 


36738 


40052 


8 


25219 


29402 


33242 


36795 


40106 


9 


25292 


29468 


33303 


36852 


40159 


10 


9.25365 


9.29535 


9.33365 


9.36909 


9.40212 


11 


25437 


29601 


33426 


36966 


40266 


12 


25510 


29668 


33487 


37023 


40319 


13 


25582 


29734 


33548 


37080 


40372 


14 


25655 


29800 


33609 


37137 


40425 


16 


9.25727 


9.29866 


9.33670 


9.37193 


9.40478 


16 


25799 


29932 


33731 


37250 


40531 


17 


25871 


29998 


33792 


37306 


40584 


18 


25943 


30064 


33853 


37363 


40636 


19 


26015 


30130 


33913 


37419 


40689 


20 


9.26086 


9.30195 


9.33974 


9.37476 


9.40742 


21 


26158 


30261 


34034 


37532 


40795 


22 


26229 


30326 


34095 


37588 


40847 


23 


26301 


30391 


34155 


37644 


40900 


24 


26372 


30457 


34215 


37700 


40952 


25 


9.26443 


9.30522 


9.34276 


9.37756 


9.41005 


26 


26514 


30587 


34336 


37812 


41057 


27 


26585 


30652 


34396 


37868 


41109 


28 


26655 


30717 


34456 


37924 


41161 


29 


26726 


30782 


34516 


37980 


41214 


30 


9.26797 


9.30846 


9.34576 


9.38035 


9.41266 


31 


26867 


30911 


34635 


38091 


41318 


32 


26937 


30975 


34695 


38147 


41370 


33 


27008 


31040 


34755 


38202 


41422 


34 


27078 


31104 


34814 


38257 


41474 


35 


9.27148 


9.31168 


9.34874 


9.38313 


9.41526 


36 


27218 


31233 


34933 


38368 


41578 


37 


27288 


31297 


34992 


38423 


41629 


38 


27357 


31361 


35051 


38479 


41681 


39 


27427 


31425 


35111 


38534 


41733 


40 


9.27496 


9.31489 


9.35170 


9.38589 


9.41784 


41 


27566 


31552 


35229 


38644 


41836 


43 


27635 


31616 


35288 


38699 


41887 


43 


27704 


31679 


35347 


38754 


41939 


44 


27773 


31743 


35405 


38808 


41990 


45 


9.27842 


9.31806 


9.35464 


9.38863 


9.42041 


46 


27911 


31870 




38918 


42093 


47 


27980 


31933 


35581 


38972 


42144 


48 


28049 


31996 


35640 


39027 


42195 


49 


28117 


32059 


35698 


39082 


42246 


50 


9.28186 


9.32122 


9.35757 


9.39136 


9.42297 


51 


28254 


32185 


35815 


39190 


42348 


63 


28323 


32248 


35873 


39245 


42399 


53 


28391 


32311 


35931 


39299 


42450 


64 


28459 


32373 


35989 


39353 


42501 


55 


9.28527 


9.32436 


9.36047 


9.39407 


9.42552 


66 


28595 


32498 


36105 


39461 


42603 


57 


28662 


32561 


36163 


39515 


42653 


68 


28730 


32623 


36221 


39569 


42704 


59 


2S798 


32685 


36279 


39623 


42755 


60 


9.28865 


9.32747 


9.36336 


9.39677 


9. 42805 




79° 


78° 


77° 


76° 


75° 


Cot 


100° 


101° 


102° 


103° 


104° 



30tj 



^^ 



TANGENTS AND COTANGENTS 






164«* 


163° 


162° 


161° 


160° 


Tan 


15° 


16° 


17° 


18° 


19° 




9.42805 


9.45750 


9.48534 


9.51178 


9.53697 


6(y 


42856 


45797 


48579 


51221 


53738 


59 


42906 


45845 


48624 


51264 


53779 


58 


42957 


45892 


48669 


51306 


53820 


57 


43007 


45940 


48714 


51349 


53861 


56 


9.43057 


9.45987 


9.48759 


9.51392 


9.53902 


55 


43108 


46035 


48804 


51435 


53943 


54 


43158 


46082 


48849 


51478 


53984 


53 


43208 


46130 


48894 


51520 


54025 


53 


43258 


46177 


48939 


51563 


54065 


51 


9.43308 


9.46224 


9.48984 


9.51606 


9.54106 


50 


43358 


46271 


49029 


51648 


54147 


49 


43408 


46319 


49073 


51691 


54187 


48 


43458 


46366 


49118 


51734 


54228 


47 


43508 


46413 


49163 


51776 


54269 


46 


9.43558 


9.46460 


9.49207 


9.51819 


9.54309 


45 


43607 


46507 


49252 


51861 


54350 


44 


43657 


46554 


49296 


51903 


54390 


43 


43707 


46601 


49341 


51946 


54431 


43 


43756 


46648 


49385 


51988 


54471 


41 


9.43806 


9.46694 


9.49430 


9.52031 


9.54512 


40 


43855 


46741 


49474 


52073 


64552 


39 


43905 


46788 


49519 


52115 


54593 


38 


43954 


46835 


49563 


52157 


54633 


37 


44004 


46881 


49607 


52200 


54673 


36 


9.44053 


9.46928 


9.49652 


9.52242 


9.54714 


35 


44102 


46975 


49696 


52284 


54754 


34 


44151 


47021 


49740 


52326 


54794 


33 


44201 


47068 


49784 


52368 


54835 


33 


44250 


47114 


49828 


52410 


54875 


31 


9.44299 


9.47160 


9.49872 


9.52452 


9.54915 


30 


44348 


47207 


49916 


52494 


54955 


39 


44397 


47253 


49960 


52536 


54995 


38 


44446 


47299 


50004 


52578 


55035 


27 


44495 


47346 


50048 


52620 


55075 


26 


9.44544 


9.47392 


9.50092 


9.52661 


9.55115 


35 


44592 


47438 


50136 


52703 


55155 


34 


44641 


47484 


50180 


52745 


55195 


23 


44690 


47530 


50223 


52787 


55235 


22 


44738 


47576 


50267 


52829 


55275 


21 


9.44787 


9.47622 


9.50311 


9.52870 


9.55315 


20 


44836 


47668 


50355 


52912 


55355 


19 


44884 


47714 


50398 


52953 


55395 


18 


44933 


47760 


50442 


52995 


55434 


17 


44981 


47806 


50485 


53037 


55474 


16 


9.45029 


9.47852 


9.50529 


9.53078 


9.55514 


15 


45078 


47897 


50572 


53120 


55554 


14 


45126 


47943 


50616 


53161 


55593 


13 


45174 


47989 


50659 


53202 


55633 


12 


45222 


48035 


50703 


53244 


55673 


11 


9.45271 


9.48080 


9.50746 


9.53285 


9.55712 


10 


45319 


48126 


50789 


53327 


55752 


9 


45367 


48171 


50833 


53368 


55791 


8 


45415 


48217 


50876 


53409 


55831 


7 


45463 


48262 


50919 


53450 


55870 


6 


9.45511 


9.48307 


9.50962 


9.53492 


9.55910 


5 


45559 


48353 


51005 


53533 


55949 


4 


45606 


48398 


51048 


53574 


55989 


3 


45654 


48443 


51092 


53615 


56028 


3 


45702 


48489 


51135 


53656 


56067 


1 


9.45750 


9.48534 


9.51178 


9.53697 


9.56107 





74° 


73° 


72° 


71° 


70° 


Cot 


105° 


106° 


107° 


108° 


109° 





207 



m. LOGARITHMIC 





159° 


158° 


157° 


156° 


155° 


Tan 


30° 


31° 


33° 


33° 


34° 


0' 


9.56107 


9.58418 


9.60641 


9.62785 


9.64858 


1 


56146 


58455 


60677 


62820 


64892 


2 


56185 


58493 


60714 


62855 


64926 


3 


56224 


58531 


60750 


62890 


64960 


4 


56264 


58569 


60786 


62926 


64994 


6 


9.56303 


9.58605 


9.60823 


9.62961 


9.65028 


6 


56342 


58644 


60859 


62996 


65062 


7 


56381 


58681 


60895 


63031 


65096 


8 


56420 


58719 


60931 


63066 


65130 


9 


56459 


58757 


60967 


63101 


65164 


10 


9.56498 


9.58794 


9.61004 


9.63135 


9.65197 


11 


56537 


58832 


61040 


63170 


65231 


13 


56576 


58869 


61076 


63205 


65265 


13 


56615 


58907 


61112 


63240 


65299 


14 


56654 


58944 


61148 


63275 


65333 


15 


9.56693 


9.58981 


9.61184 


9.63310 


9.65366 


16 


56732 


59019 


61220 


63345 


65400 


17 


56771 


59056 


61256 


63379 


65434 


18 


56810 


59094 


61292 


63414 


65467 


19 


56849 


59131 


61328 


63449 


65501 


20 


9.56887 


9.59168 


9.61364 


9.63484 


9.65535 


21 


56926 


59205 


61400 


63519 


65568 


23 


56965 


59243 


61436 


63553 


65692 


23 


57004 


59280 


61472 


63588 


65636 


34 


57042 


59317 


61508 


63623 


65669 


35 


9.75081 


9.59354 


9.61544 


9.63657 


9.65703 


36 


57120 


59391 


61579 


63692 


65736 


37 


57158 


59429 


61615 


63726 


65770 


38 


57197 


59466 


61651 


63761 


65803 


39 


57235 


59503 


61687 


63796 


65837 


30 


9.57274 


9.59540 


9.61722 


9.63830 


9.65870 


31 


57312 


59577 


61758 


63865 


65904 


32 


57351 


59614 


61794 


63899 


65937 


33 


57389 


59651 


61830 


63934 


65971 


34 


57428 


59688 


61865 


63968 


66004 


35 


9.57466 


9.59725 


9.61901 


9.64003 


9.66038 


36 


57504 


59762 


61936 


64037 


66071 


37 


57543 


59799 


61972 


64072 


66104 


38 


57581 


59835 


62008 


64106 


66138 


39 


57619 


59872 


62043 


64140 


66171 


40 


9.57658 


9.59909 


9.62079 


9.64175 


9.66204 


41 


57696 


59946 


62114 


64209 


66238 


43 


57734 


59983 


62150 


64243 


66271 


43 


57772 


60019 


62185 


64278 


66304 


44 


57810 


60056 


62221 


64312 


66337 


45 


9.57849 


9.60093 


9.62256 


9.64346 


9.66371 


46 


57887 


60130 


62292 


64381 


66404 


47 


57925 


60166 


62327 


64415 


66437 


48 


57963 


60203 


62362 


64449 


66470 


49 


58001 


60240 


62398 


64483 


66503 


50 


9.58039 


9.60276 


9.62433 


9.64517 


9.66537 


51 


58077 


60313 


62468 


64552 


66570 


53 


58115 


60349 


62504 


64586 


66603 


53 


58153 


60386 


62539 


64620 


66636 


54 


58191 


60422 


62574 


64654 


66669 


55 


9.58229 


9.60459 


9.62609 


9.64688- 


9.66702 


56 


58267 


60495 


02645 


64722 


66735 


57 


58304 


60532 


62680 


64756 


66768 


58 


58342 


60568 


62715 


64790 


66801 


59 


58380 


60605 


62750 


64824 


66834 


60 


9.58418 


9.00641 


9.62785 


9.64858 


9.66867 




69° 


68° 


67° 


66° 


65° 


Cot 


110° 


111° 


112° 


113^ 


114° 



208 



TANGENTS AND COTANGENTS 



154° 

35° 


153° 


152° 


151° 


150° 


Tan 


26° 


27° 


28° 


29° 




9.66867 


9.68818 


9.70717 


9.72567 


9.74375 


60^ 


66900 


68850 


70748 


72598 


74405 


59 


66933 


68882 


70779 


72628 


74435 


58 


66966 


68914 


70810 


72659 


74465 


57 


66999 


68946 


70841 


72689 


74494 


56 


9.67032 


9.68978 


9.70873 


9.72720 


9.74524 


55 


67065 


69010 


70904 


72750 


74554 


54 


67098 


69042 


70935 


72780 


74583 


53 


67131 


69074 


70966 


72811 


74613 


53 


67163 


69106 


70997 


72841 


74643 


51 


9.67196 


9.69138 


9.71028 


9.72872 


9.74673 


50 


67229 


69170 


71059 


72902 


74702 


49 


€7262 


69202 


71090 


72932 


74732 


48 


67295 


69234 


71121 


72963 


74762 


47 


67327 


69266 


71153 


72993 


74791 


46 


9.67360 


9.69298 


9.71184 


9.73023 


9.74821 


45 


67393 


69329 


71215 


73054 


74851 


44 


67426 


69361 


71246 


73084 


74880 


43 


67458 


69393 


71277 


73114 


74910 


43 


67491 


69425 


71308 


73144 


74939 


41 


9.67524 


9.69457 


9.71339 


9.73175 


9.74969 


40 


67556 


69488 


71370 


73205 


74998 


39 


67589 


69520 


71401 


73235 


75028 


38 


67622 


69552 


71431 


73265 


75058 


37 


67654 


69584 


71462 


73295 


75087 


36 


9.67687 


9.69615 


9.71493 


9.73326 


9.75117 


35 


67719 


69647 


71524 


73356 


75146 


34 


67752 


69679 


71555 


73386 


75176 


33 


67785 


69710 


71586 


73416 


75205 


33 


67817 


69742 


71617 


73446 


75235 


31 


9.67850 


9.69774 


9.71648 


9.73476 


9.75264 


30 


67882 


69805 


71679 


73507 


75294 


39 


67915 


69837 


71709 


73537 


75323 


38 


67947 


69868 


71740 


73567 


75353 


37 


67980 


69900 


71771 


73597 


75382 


38 


9.68012 


9.69932 


9.71802 


9.73627 


9.75411 


35 


68044 


69963 


71833 


73657 


75441 


34 


68077 


69995 


71863 


73687 


75470 


33 


68109 


70026 


71894 


73717 


75500 


33 


68142 


70058 


71925 


73747 


75529 


31 


9.68174 


9.70089 


9.71955 


9.73777 


9.75558 


30 


68206 


70121 


71986 


73807 


75588 


19 


68239 


70152 


72017 


73837 


75617 


18 


68271 


70184 


72048 


73867 


75647 


17 


68303 


70215 


72078 


73897 


75676 


16 


9.68336 


9.70247 


9.72109 


9.73927 


9.75705 


15 


68368 


70278 


72140 


73957 


75735 


14 


68400 


70309 


72170 


73987 


75764 


13 


68432 


70341 


72201 


74017 


75793 


13 


68465 


70372 


72231 


74047 


75822 


11 


9.68497 


9.70404 


9.72262 


9.74077 


9.75852 


10 


68529 


70435 


72293 


74107 


75881 


9 


68561 


70466 


72323 


74137 


75910 


8 


68593 


70498 


72354 


74166 


75939 


7 


68626 


70529 


72384 


74196 


75969 


6 


9.68658 


9.70560 


9.72415 


9.74226 


9.75998 


5 


68690 


70592 


72445 


74256 


76027 


4 


68722 


70623 


72476 


74286 


76056 


3 


68754 


70654 


72506 


74316 


76086 


3 


68786 


70685 


72537 


74345 


76115 


1 


9.68818 


9.70717 


9.72567 


9.74375 


9.76144 





64° 


63° 


63° 


61° 


60° 


Cot 


115° 


116° 


117° 


118° 


119° 





209 



in. LOGARITHMIC 





149° 


148° 


147° 


146° 


145° 


Tan 


30° 


31° 


33° 


33° 


34° 


0' 


9.76144 


9.77877 


9.79579 


9.81252 


9.82899 


1 


76173 


77906 


79607 


81279 


82926 


2 


76202 


77935 


79635 


81307 


82953 


3 


76231 


77963 


79663 


81335 


82980 


4 


76261 


77992 


79691 


81362 


83008 


5 


9.76290 


9.78020 


9.79719 


9.81390 


9.83035 


6 


76319 


78049 


79747 


81418 


83062 


7 


76348 


78077 


79776 


81445 


83089 


8 


76377 


78106 


79804 


81473 


83117 


9 


76406 


78135 


79832 


81500 


83144 


10 


9.76435 


9.78163 


9.79860 


9.81528 


9.83171 


11 


76464 


78192 


79888 


81556 


83198 


13 


76493 


78220 


79916 


81583 


83225 


13 


76522 


78249 


79944 


81611 


83252 


14 


76551 


78277 


79972 


81638 


83280 


15 


9.76580 


9.78306 


9.80000 


9.81666 


9.83307 


16 


76609 


78334 


80028 


81693 


83334 


17 


76639 


78363 


80056 


81721 


83361 


18 


76668 


78391 


80084 


81748 


83388 


19 


76697 


78419 


80112 


81776 


83415 


30 


9.76725 


9.78448 


9.80140 


9.81803 


9.83442 


31 


76754 


78476 


80168 


81831 


83470 


33 


76783 


78505 


80195 


81858 


83497 


33 


76812 


78533 


80223 


81886 


83524 


34 


76841 


78562 


80251 


81913 


83551 


35 


9.76870 


9.78590 


9.80279 


9.81941 


9.83578 


36 


76899 


78618 


80307 


81968 


83605 


37 


76928 


78647 


80335 


81996 


83632 


38 


76957 


78675 


80363 


82023 


83659 


39 


76986 


78704 


80391 


82051 


83683 


30 


9.77015 


9.78732 


9.80419 


9.82078 


9.83713 


31 


77044 


78760 


80447 


82106 


83740 


33 


77073 


78789 


80474 


82133 


83768 


33 


77101 


78817 


80502 


82161 


83795 


34 


77130 


78845 


80530 


82188 


83S22 


35 


9.77159 


9.78874 


9.80558 


9.82215 


9.83849 


36 


77188 


78902 


80586 


82243 


83876 


37 


77217 


78930 


80614 


82270 


83903 


38 


77246 


78959 


80642 


82298 


83930 


39 


77274 


78987 


80669 


82325 


83957 


40 


9.77303 


9.79015 


9.80697 


9.82352 


9.83984 


41 


77332 


79043 


80725 


82380 


84011 


43 


77361 


79072 


80753 


82407 


84038 


43 


77390 




80781 


82435 


84065 


44 


77418 


79128 


80808 


82462 


84092 


45 


9.77447 


9.79156 


9.80836 


9.82489 


9.84119 


46 


77476 


79185 


80864 


82517 


84146 


47 


77505 


79213 


80892 


82544 


84173 


48 


77533 


79241 


80919 


82571 


84200 


49 


77562 


79269 


80947 


82599 


84227 


50 


9.77591 


9.79297 


9.80975 


9.82626 


9.84254 


51 


77619 


79326 


81003 


82653 


84280 


53 


77648 


79354 


81030 


S2681 


84307 


53 


77677 


79382 


81058 


82708 


84334 


54 


77706 


79410 


81086 


82735 


84361 


55 


9.77734 


9.79438 


9.81113 


9.82762 


9.84388 


56 


77763 


79466 


81141 


82790 


84415 


57 


77791 


79495 


81169 


82817 


84442 


58 


77820 


79523 


81196 


82844 


84469 


59 


77849 


79551 


81224 


82871 


84496 


60 


9.77877 


9.79579 


9.81252 


9.82899 


9.84523 




59° 


58° 


57° 


56° 


55° 

124° 


Cot 


120° 


121° 


122° 


123° 



210 





TANGENTS AND 


COTANGENTS 




^ 




144° 


143° 


142° 


141° 


140° 


Tan 


35° 


36° 


37° 


38° 


39° 




9.84523 


9.86126 


9.87711 


9.89281 


9.90837 


6(y 




84550 


86153 


87738 


89307 


90863 


59 




84576 


86179 


87764 


89333 


90889 


58 




84603 


86206 


87790 


89359 


90914 


57 




84630 


86232 


87817 


89385 


90940 


56 




9.84657 


9.86259 


9.87843 


9.89411 


9.90966 


55 




84684 


862S5 


87869 


89437 


90992 


54 




84711 


86312 


87895 


89463 


91018 


53 




84738 


86338 


87922 


89489 


91043 


52 




84764 


86365 


87948 


89515 


91069 


51 




9.84791 


9.86392 


9.87974 


9.89541 


9.91095 


50 




84818 


86418 


88000 


89567 


91121 


49 




84845 


86445 


88027 


89593 


91147 


48 




84872 


86471 


88053 


89619 


91172 


47 




84899 


86498 


88079 


89645 


91198 


46 




9.84925 


9.86524 


9.88105 


9.89671 


9.91224 


45 




84952 


86551 


88131 


89697 


91250 


44 




84979 


86577 


88158 


89723 


91276 


43 




85006 


86603 


88184 


89749 


91301 


43 


1 


85033 


86630 


88210 


89775 


91327 


41 




9.85059 


9.86656 


9.88236 


9.89801 


9.91353 


40 




85086 


86683 


88262 


89827 


91379 


39 




85113 


86709 


88289 


89853 


91404 


38 




85140 


86736 


88315 


89879 


91430 


37 




85166 


86762 


88341 


89905 


91456 


36 


1 


9.85193 


9.86789 


9.88367 


9.89931 


9.91482 


35 


1 


85220 


86815 


88393 


89957 


91507 


34 




85247 


86842 


88420 


89983 


91533 


33 




85273 


86868 


88446 


90009 


91559 


33 




85300 


86894 


88472 


90035 


91585 


31 




9.85327 


9.86921 


9.88498 


9.90061 


9.91610 


30 




85354 


86947 


88524 


90086 


91636 


39 




85380 


86974 


88550 


90112 


91662 


38 




85407 


87000 


88577 


90138 


91688 


37 




85434 


87027 


88603 


90164 


91713 


36 




9.85460 


9.87053 


9.88629 


9.90190 


9.91739 


35 




85487 


87079 


88655 


90216 


91765 


34 




85514 


87106 


88681 


90242 


91791 


33 




85540 


87132 


88707 


90268 


91816 


33 




85567 


87158 


88733 


90294 


91842 


31 




9.85594 


9.87185 


9.88759 


9.90320 


9.91868 


30 


1 


85620 


87211 


88786 


90346 


91893 


19 




85647 


87238 


88812 


90371 


91919 


18 




85674 


87264 


88838 


90379 


91945 


17 




85700 


87290 


88864 


90423 


91971 


16 




9.85727 


9.87317 


9.88890 


9.90449 


9.91996 


15 




85754 


87343 


88916 


90475 


92022 


14 'j 




85780 


87369 


88942 


90501 


92048 


13 i 


1 


85807 


87396 


88968 


90527 


92073 


13 1 


i 


85834 


87422 


88994 


90553 


92099 


11 1 




9.85860 


9.87448 


9.89020 


9.90578 


9.92125 


10 ll 




85887 


87475 


89046 


90604 


92150 


9 fl 


85913 


87501 


89073 


90630 


92176 


8 H 




85940 


87527 


89099 


90656 


92202 


7 




85967 


87554 


89125 


90682 


92227 


^ 




9.85993 


9.87580 


9.89151 


9.90708 


9.92253 


5 




86020 


87606 


89177 


90734 


92279 


4 y 




86046 


87633 


89203 


90759 


92304 


3 H 




86073 


87659 


89229 


90785 


92330 


3 n 




86100 


87685 


89255 


90811 


92356 


1 




9.86126 


9.87711 


9.89281 


9.90837 


9.92381 


II 

Cot ji II 


54° 


53° 


53° 


51° 


50° 


125° 


126° 


127° 128° 1 


129° 


ii 



211 



III. LOGARITHMIC 





139° 


138° 


137° 


136° 


135° 


Tan 


40° 


41° 


43° 


43° 


44° 


0' 


9.92381 


9.93916 


9.95444 


9.96966 


9.98484 


1 


92407 


93942 


95469 


96991 


98509 


2 


92433 


93967 


95495 


97016 


98534 


3 


92458 


93993 


95520 


97042 


98560 


4 


92484 


94018 


95545 


97067 


98585 


5 


9.92510 


9.94044 


9.95571 


9.97092 


9.98610 


6 


92535 


94069 


95506 


97118 


98635 


7 


92561 


94095 


95622 


97143 


98661 


8 


92587 


94120 


95647 


97168 


98686 


9 


92612 


94146 


95672 


97193 


98711 


10 


9.92638 


9.94171 


9.95698 


9.97219 


9.98737 


11 


92663 


94197 


95723 


97244 


98762 


12 


92689 


94222 


95748 


97269 


98787 


13 


92715 


94248 


95774 


97295 


98812 


14 


92740 


94273 


95799 


97320 


98838 


15 


9.92766 


9.94299 


9.95825 


9.97345 


9.98863 


16 


92792 


94324 


95850 


97371 


98888 


17 


92817 


94350 


95875 


97396 


98913 


18 


92843 


94375 


95901 


97421 


98939 


19 


92868 


94401 


95926 


97447 


98964 


30 


9.92894 


9.94426 


9.95952 


9.97472 


9.98989 


31 


92920 


94452 


95977 


97497 


99015 


33 


92945 


94477 


96002 


97523 


99040 


33 


92971 


94503 


96028 


97548 


99065 


34 


92996 


94528 


96053 


97573 


99090 


35 


9.93022 


9.94554 


9.96078 


9.97598 


9.99116 


36 


93048 


94579 


96104 


97624 


99141 


37 


93073 


94604 


96129 


97649 


99166 


38 


93099 


94630 


96155 


97674 


99191 


39 


93124 


94655 


96180 


97700 


99217 


30 


9.93150 


9.94681 


9.96205 


9.97725 


9.99242 


31 


93175 


94706 


96231 


97750 


99267 


33 


93201 


94732 


96256 


97776 


99293 


33 


93227 


94757 


96281 


97801 


99318 


34 


93252 


94783 


96307 


97826 


99343 


35 


9.93278 


9.94808 


9.96332 


9.97851 


9.99368 


36 


93303 


94834 


96357 


97877 


99394 


37 


93329 


94859 


96383 


97902 


99419 


38 


93354 


94884 


96408 


97927 


99444 


39 


93380 


94910 


96433 


97953 


99469 


40 


9.93406 


9.94935 


9.96459 


9.97978 


9.99495 


41 


93431 


94961 


96484 


98003 


99520 


43 


93457 


94986 


9S510 


98029 


99545 


43 


93482 


95012 


96535 


98054 


99570 


44 


93508 


95037 


96560 


98079 


99596 


45 


9.93533 


9.95062 


9.96586 


9.98104 


9.99621 


46 


93559 


95088 


96611 


98130 


99646 


47 


93584 


95113 


96636 


98155 


99672 


48 


93610 


95139 


96662 


98180 


99697 


49 


93636 


95164 


96687 


98206 


99722 


50 


9.93661 


9.95190 


9.96712 


9.98231 


9.99747 


51 


93687 


95215 


96738 


98256 


99773 


53 


93712 


95240 


96763 


98281 


99798 


53 


93738 


95266 


96788 


98307 


99823 


54 


93763 


95291 


96814 


98332 


99848 


55 


9.93789 


9.95317 


9.96839 


9.98357 


9.99874 


56 


93814 


95342 


96864 


98383 


99899 


57 


93840 


95368 


96890 


98408 


99924 


58 


93865 


95393 


96915 


98433 


99949 


59 


93891 


95418 


96940 


98458 


99975 


60 


9.93916 


9.95444 


9.96906 


9.98484 


0.00000 




49° 


48° 


47° 


46° 


45° 


Cot 


130° 


131° 


132° 


133° 


134° 



212 



TANGENTS AND COTANGENTS 



134° 


133° 
46° 


132° 


131° 


130° 


Tan 


45° 


47° 


48° 


49° 




0.00000 


0.01516 


0.03034 


0.04556 


0.06084 


6(y 


00025 


01542 


03060 


04582 


06109 


59 


00051 


01567 


03085 


04607 


06135 


58 


00076 


01592 


03110 


04632 


06160 


57 


00101 


01617 


03136 


04658 


06186 


56 


0.00126 


0.01643 


0.03161 


0.04683 


0.06211 


55 


00152 


01668 


03186 


04709 


06237 


54 


00177 


01693 


03212 


04734 


06262 


53 


00202 


01719 


03237 


04760 


06288 


53 


00227 


01744 


03262 


04785 


06313 


51 


0.00253 


0.01769 


0.03288 


0.04810 


0.06339 


50 


00278 


01794 


03313 


04836 


06364 


49 


00303 


01820 


03338 


04861 


06390 


48 


00328 


01845 


03364 


04887 


06416 


47 


00354 


01870 


03389 


04912 


06441 


46 


0.00379 


0.01896 


0.03414 


0.04938 


0.06467 


45 


00404 


01921 


03440 


04963 


06492 


44 


00430 


01946 


03465 


04988 


06518 


43 


00455 


01971 


03490 


05014 


06543 


43 


00480 


01997 


03516 


05039 


06569 


41 


0.00505 


0.02022 


0.03541 


0.05065 


0.06594 


40 


00531 


02047 


03567 


05090 


06620 


39 


00556 


02073 


03592 


05116 


06646 


38 


00581 


02098 


03617 


05141 


06671 


37 


00606 


02123 


03643 


05166 


06697 


36 


0.00632 


0.02149 


0.03668 


0.05192 


0.06722 


35 


00657 


02174 


03693 


05217 


06748 


34 


00682 


02199 


03719 


05243 


06773 


33 


00707 


02224 


03744 


05268 


06799 


32 


00733 


02250 


03769 


05294 


06825 


31 


0.00758 


0.02275 


0.03795 


0.05319 


0.06850 


30 


00783 


02300 


03820 


05345 


06876 


29 


00809 


02326 


03845 


05370 


06901 


28 


00834 


02351 


03871 


05396 


06927 


27 


00859 


02376 


03896 


05421 


06952 


26 


0.00884 


0.02402 


0.03922 


0.05446 


0.06978 


25 


00910 


02427 


03947 


05472 


07004 


24 


00935 


02452 


03972 


05497 


07029 


23 


00960 


02477 


03998 


05523 


07055 


23 


00985 


02503 


04023 


05548 


07080 


21 


0.01011 


0.02528 


0.04048 


0.05574 


0.07106 


20 


01036 


02553 


04074 


05599 


07132 


19 


01061 


02579 


04099 


05625 


07157 


18 


01087 


02604 


04125 


05650 


07183 


17 


01112 


02629 


04150 


05676 


07208 


16 


0.01137 


0.02655 


0.04175 


0.05701 


0.07234 


15 


01162 


02680 


04201 


05727 


07260 


14 


01188 


02705 


04226 


05752 


07285 


13 


01213 


02731 


04252 


05778 


07311 


12 


01238 


02756 


04277 


05803 


07337 


11 


0.01263 


0.02781 


0.04302 


0.05829 


0.07362 


10 


01289 


02807 


04328 


05854 


07388 


9 


01314 


02832 


04353 


05880 


07413 


8 


01339 


02857 


04378 


05905 


07439 


7 


01365 


02882 


04404 


05931 


07465 


6 


0.01390 


0.02908 


0.04429 


0.05956 


0.07490 


5 


01415 


02933 


04455 


05982 


07516 


4 


01440 


02958 


04480 


06007 


07542 


3 


01466 


02984 


04505 


06033 


07567 
07593 


2 


01491 


03009 


04531 


06058 


1 


0.01516 


0.03034 


0.04556 


0.06084 


0.07619 





44° 


43° 


43° 


41° 


40° 


Cot 


135° 


136° 


137° 


138° 


139° 





213 



III. LOGARITHMIC 





129° 


128° 


127° 


126° 


125° 


Tan 


50° 


51° 


52° 


53° 


54° 


0' 


0.07619 


0.09163 


0.10719 


0.12289 


0.13874 


1 


07644 


09189 


10745 


12315 


13900 


2 


07670 


09215 


10771 


12341 


13927 


3 


07696 


09241 


10797 


12367 


13954 


4 


07721 


09266 


10823 


12394 


13980 


5 


0.07747 


0.09292 


0.10849 


0. 12420 


0.14007 


6 


07773 


09318 


10875 


12446 


14033 


7 


07798 


09344 


10901 


12473 


14060 


8 


07824 


09370 


10927 


12499 


14087 


9 


07850 


09396 


10954 


12525 


14113 


10 


0.07875 


0.09422 


0.10980 


0. 12552 


0.14140 


11 


07901 


09447 


11006 


12578 


14166 


13 


07927 


09473 


11032 


12604 


14193 


13 


07952 


09499 


11058 


12631 


14220 


14 


07978 


09525 


11084 


12657 


14246 


15 


0.08004 


0.09551 


0.11110 


0.12683 


0.14273 


16 


08029 


09577 


11136 


12710 


14300 


17 


08055 


09603 


11162 


12736 


14326 


18 


08081 


09629 


11188 


12762 


14353 


19 


08107 


09654 


11214 


12789 


14380 


20 


0.08132 


0.09680 


0.11241 


0.12815 


0.14406 


21 


08158 


09706 


11267 


12842 


14433 


22 


08184 


09732 


11293 


12868 


14460 


23 


08209 


09758 


11319 


12894 


14486 


24 


08235 


09784 


11345 


12921 


14513 


25 


0.08261 


0.09810 


0.11371 


0. 12947 


0.14540 


26 


08287 


09836 


11397 


12973 


14566 


27 


08312 


09862 


11423 


13000 


14593 


28 


08338 


09888 


11450 


13026 


14620 


29 


08364 


09914 


11476 


13053 


14646 


30 


0.08390 


0.09939 


0.11502 


0. 13079 


0.14673 


31 


08415 


09965 


11528 


13106 


14700 


32 


08441 


09991 


11554 


13132 


14727 


33 


08467 


10017 


11580 


13158 


14753 


34 


08493 


10043 


11607 


13185 


14780 


35 


0.08518 


0.10069 


0.11633 


0.13211 


0.14807 


36 


08544 


10095 


11659 


13238 


14834 


37 


08570 


10121 


11685 


13264 


14860 


38 


08596 


10147 


11711 


13291 


14887 


39 


08621 


10173 


11738 


13317 


14914 


40 


0.08647 


0.10199 


0.11764 


0.13344 


0.14941 


41 


08673 


10225 


11790 


13370 


14967 


42 


08699 


10251 


11816 


13397 


14994 


43 


08724 


10277 


11842 


13423 


15021 


44 


08750 


10303 


11869 


13449 


15048 


45 


0.08776 


0.10329 


0.11895 


0. 13476 


0.15075 


46 


08802 


10355 


11921 


13502 


15101 


47 


08828 


10381 


11947 


13529 


15128 


4S 


08853 


10407 


11973 


13555 


15155 


49 


08879 


10433 


12000 


13582 


15182 


50 


0.0S905 


0.10459 


0.12026 


0. 13608 


0. 15209 


51 


08931 


10485 


12052 


13635 


15236 


52 


08957 


10511 


12078 


13662 


15262 


53 


08982 


10537 


12105 


13688 


15289 


54 


0900S 


10563 


12131 


13715 


15316 


55 


0.09034 


0.10589 


0.12157 


0.13741 


0. 15343 


56 


09060 


10615 


12183 


13768 


15370 


57 


09086 


10641 


12210 


13794 


15397 


58 


09111 


10667 


12236 


13821 


15424 


59 


09137 


10693 


12262 


13847 


15450 


60 


0.09163 


0.10719 


0. 12289 


0.13874 


0.15477 




39° 


38° 


37° 


36° 


35° 


Cot 


140° 


141° 


142° 


143° 


144° 



214 



TANGEHTS AND COTANGENTS 



124° 1 


123° 


122° 


121° 


120° 


Tan 


55° 


56° 


57° 


58° 


59° 




0.15477 


0.17101 


0.18748 


0.20421 


0.22123 


60' 


15504 


17129 


18776 


20449 


22151 


59 


15531 


17156 


18804 


20477 


22180 


58 


15558 


17183 


18831 


20505 


22209 


57 


15585 


17210 


18859 


20534 


22237 


56 


0.15612 


0.17238 


0.18887 


0.20562 


0.22266 


55 


15639 


17265 


18914 


20590 


22294 


54 


15666 


17292 


18942 


20618 


22323 


53 


15693 


17319 


18970 


20646 


22352 


53 


15720 


17347 


18997 


20674 


22381 


51 


0.15746 


0.17374 


0.19025 


0.20703 


0.22409 


50 


15773 


17401 


19053 


20731 


22438 


49 


15800 


17429 


19081 


20759 


23467 


48 


15827 


17456 


19108 


20787 


23495 


47 


15854 


17483 


19136 


20815 


23524 


46 


0.15881 


0.17511 


0.19164 


0.20844 


0.23553 


45 


15908 


17538 


19192 


20872 


23582 


44 


15935 


17565 


19219 


20900 


23610 


43 


15962 


17593 


19247 


20928 


23639 


43 


15989 


17620 


19275 


20957 


22668 


41 


0. 16016 


0.1764S 


0.19303 


0.20985 


0.22697 


40 


16043 


17675 


19331 


21013 


22726 


39 


16070 


17702 


19358 


21041 


22754 


38 


16097 


17730 


19386 


21070 


22783 


37 


16124 


17757 


19414 


21098 


22812 


36 


0.16151 


0. 17785 


0.19442 


0.21126 


0.22841 


35 


16178 


17812 


19470 


21155 


22870 


34 


16205 


17839 


19498 


21183 


22899 


33 


16232 


17867 


19526 


21211 


22927 


33 


16260 


17894 


19553 


21240 


22950 


31 


0.16287 


0.17922 


0.19581 


0.21268 


0.22985 


30 


16314 


17949 


19609 


21296 


23014 


29 


16341 


17977 


19637 


21325 


23043 


28 


16368 


18004 


19665 


21353 


23072 


27 


16395 


18032 


19693 


21382 


23101 


26 


0.16422 


0.18059 


0.19721 


0.21410 


0.23130 


25 


16449 


18087 


19749 


21438 


23159 


24 


16476 


18114 


19777 


21467 


23188 


33 


16503 


18142 


19805 


21495 


23217 


33 


16530 


18169 


19832 


21524 


23246 


31 


0.16558 


0.18197 


0.19860 


0.21552 


0.23275 


30 


16585 


18224 


19888 


21581 


23303 


19 


16612 


18252 


19916 


21609 


23332 


18 


16639 


18279 


19944 


21637 


23361 


17 


16666 


18307 


19972 


21666 


23391 


16 


0.16693 


0.18334 


0.20000 


0.21694 


0.23420 


15 


16720 


18362 


20028 


21723 


23449 


14 


16748 


18389 


20056 


21751 


23478 


13 


16775 


18417 


20084 


21780 


23507 


13 


16802 


18444 


20112 


21808 


23536 


11 


0.16829 


0. 18472 


0.20140 


0.21837 


0.23565 


10 


16856 


18500 


20168 


21865 


23594 


9 


16883 


18527 


20196 


21894 


23623 


8 


16911 


18555 


20224 


21923 


23652 


7 


16938 


18582 


20253 


21951 


23681 


6 


0. 16965 


0.18610 


0.20281 


0.21980 


0.23710 


5 


16992 


18638 


20309 


22008 


23739 


4 


17020 


18665 


20337 


22037 


23769 


3 


17047 


18693 


20365 


22065 


23798 


2 


17074 


18721 


20393 


22094 


23827 


1 


0.17101 


0.18748 


0.20421 


0.22123 


0.23856 





34° 


33° 


32° 


31° 


30° 


Cot 


145° 


146° 


147° 


148° 


149° 





215 



III. LOGARITHMIC 





119° 


118° 


117° 


116° 


115° 


Tan 


60° 


61° 


63° 


63° 


64° 


0' 


0.23856 


0.25625 


0.27433 


0.29283 


0.31182 


1 


23885 


25655 


27463 


29315 


31214 


2 


23914 


25684 


27494 


29346 


31246 


3 


23944 


25714 


27524 


29377 


31278 


4 


23973 


25744 


27555 


29408 


31310 


5 


0.24002 


0.25774 


0.27585 


0.29440 


0.31342 


6 


24031 


25804 


27616 


29471 


31374 


7 


24061 


25834 


27646 


29502 


31407 


8 


24090 


25863 


27677 


29534 


31439 


9 


24119 


25893 


27707 


29565 


31471 


10 


0.24148 


0.25923 


0.27738 


0.29596 


0.31503 


11 


24178 


25953 


27769 


29628 


31535 


13 


24207 


25983 


27799 


29659 


31568 


13 


24236 


26013 


27830 


29691 


31600 


14 


24265 


26043 


27860 


29722 


31632 


15 


0.24295 


0.26073 


0.27891 


0.29753 


0.31664 


16 


24324 


26103 


27922 


29785 


31697 


17 


24353 


26133 


27952 


29816 


31729 


18 


24383 


26163 


27983 


29848 


31761 


19 


24412 


26193 


28014 


29879 


31794 


20 


0.24442 


0.26223 


0.28045 


0.29911 


0.31826 


31 


24471 


26253 


28075 


29942 


31858 


22 


24500 


26283 


28106 


29974 


31891 


33 


24530 


26313 


28137 


30005 


31923 


34 


24559 


26343 


28167 


30037 


31956 


35 


0.24589 


0.26373 


0.28198 


0.30068 


0.31988 


36 


24618 


26403 


28229 


30100 


32020 


37 


24647 


26433 


• 28260 


30132 


32053 


38 


24677 


26463 


28291 


30163 


32085 


39 


24706 


26493 


28321 


30195 


32118 


30 


0.24736 


0.26524 


0.28352 


0.30226 


0.32150 


31 


24765 


26554 


28383 


30258 


32183 


33 


24795 


26584 


28414 


30290 


32215 


33 


24824 


26614 


28445 


30321 


32248 


34 


24854 


26644 


28476 


30353 


32281 


35 


0.24883 


0.26674 


0.28507 


0.30385 


0.32313 


36 


24913 


26705 


28538 


30416 


32346 


37 


24942 


26735 


28569 


30448 


32378 


38 


24972 


26765 


28599 


30480 


32411 


39 


25002 


26795 


28630 


30512 


32444 


40 


0.25031 


0.26825 


0.28661 


0.30543 


0.32476 


41 


25061 


26856 


28692 


30575 


32509 


43 


25090 


26886 


38723 


30607 


32542 


43 


25120 


26916 


28754 


30639 


32574 


44 


25149 


26946 


28785 


30671 


32607 


45 


0.25179 


0.26977 


0.28816 


0.30702 


0.32640 


46 


25209 


27007 


28847 


30734 


32673 


47 


25238 


27037 


28879 


30766 


32705 


48 


25268 


27068 


28910 


30798 


32738 


49 


25298 


27098 


28941 


30830 


32771 


50 


0.25327 


0.27128 


0.28972 


0.30862 


0.32804 


51 


25357 


27159 


29003 


30894 


32837 


53 


25387 


27189 


29034 


30926 


32869 


53 


25417 


27220 


29065 


30958 


32902 


54 


25446 


27250 


29096 


30990 


32935 


55 


0.25476 


0.27280 


0.29127 


0.31022 


0.32968 


56 


25503 


27311 


29159 


31054 


33001 


57 


25535 


27341 


29190 


31086 


33034 


58 


25565 


27372 


29221 


3111S 


33067 


59 


25595 


27402 


29252 


31150 


33100 


60 


0.25625 


0.27433 


0.29283 


0.31182 


0.33133 




39° 


38° 


37° 


36° 


35° 


Cot 


150° 


151° 


152° 


153° 


154° 



210 



TANGENTS AND COTANGENTS 



114° 


113° 


112° 


111° 


110° 


Tan 


65° 


66° 


67° 


68° 


69° 




0.33133 


0.35142 


0.37215 


0.39359 


0.41582 


G(y 


33166 


35176 


37250 


39395 


41620 


59 


33199 


35210 


37285 


39432 


41658 


58 


33232 


35244 


37320 


39468 


41696 


57 


33265 


35278 


37355 


39505 


41733 


56 


0.33298 


0.35312 


0.37391 


0.39541 


0.41771 


55 


33331 


35346 


37426 


39578 


41809 


54 


33364 


35380 


37461 


39614 


41847 


53 


33397 


35414 


37496 


39651 


41885 


52 


33430 


35448 


37532 


39687 


41923 


51 


0.33463 


0.35483 


0.37567 


0.39724 


0.41961 


50 


33497 


35517 


37602 


39760 


41999 


49 


33530 


35551 


37638 


39797 


42037 


48 


33563 


35585 


37673 


39834 


42075 


47 


33596 


35619 


37708 


39870 


42113 


46 


0.33629 


0.35654 


0.37744 


0.39907 


0.42151 


45 


33663 


35688 


37779 


39944 


42190 


44 


33696 


35722 


37815 


39981 


42228 


43 


33729 


35757 


37850 


40017 


42266 


42 


33762 


35791 


37886 


40054 


42304 


41 


0.33796 


0.35S25 


0.37921 


0.40091 


0.42342 


40 


33829 


35860 


37957 


40128 


42381 


39 


33862 


35894 


37992 


40165 


42419 


38 


33896 


35928 


38028 


40210 


42457 


37 


33929 


35963 


38064 


40238 


42496 


36 


0.33962 


0.35997 


0.38099 


0.40275 


0.42534 


35 


33996 


36032 


38135 


40312 


42572 


34 


34029 


36066 


38170 


40349 


42611 


33 


34063 


36101 


38206 


40386 


42649 


32 


34096 


36135 


38242 


40423 


42688 


31 


0.34130 


0.36170 


0.38278 


0.40460 


0.42726 


30 


34163 


36204 


38313 


40497 


42765 


29 


34197 


36239 


38349 


40534 


42803 


28 


34230 


36274 


38385 


40571 


42842 


27 


34264 


36308 


38421 


40609 


42880 


26 


0.34297 


0.36343 


0.38456 


0.40646 


0.42919 


25 


34331 


36377 


38492 


40683 


42958 


24 


34364 


36412 


38528 


40720 


42996 


23 


34398 


36447 


38564 


40757 


43035 


22 


34432 


36481 


38600 


40795 


43074 


21 


0.34465 


0.36516 


0.38636 


0.40832 


0.43113 


20 


34499 


36551 


38672 


40869 


43151 


19 


34533 


36586 


38708 


40906 


43190 


18 


34566 


36621 


38744 


40944 


43229 


17 


34600 


36655 


36780 


40981 


43268 


16 


0.34634 


0.36690 


0.38816 


0.41019 


0.43307 


15 


34667 


36725 


38852 


41056 


43346 


14 


34701 


36760 


38888 


41093 


43385 


13 


34735 


36795 


38924 


41131 


43424 


12 


34769 


36830 


38960 


41168 


43463 


11 


0.34803 


0.36865 


0.38996 


0.41206 


0.43502 


10 


34836 


36899 


39033 


41243 


43541 


9 


34870 


36934 


39069 


41281 


43580 


8 


34904 


36969 


39105 


41319 


43619 


7 


34938 


37004 


39141 


41356 


43658 


6 


0.34972 


0.37039 


0.39177 


0.41394 


0.43697 


5 


35006 


37074 


39214 


41431 


43736 


4 


35040 


37110 


39250 


41469 


43776 


3 


35074 


37145 


39286 


41507 


43815 


2 


35108 


37180 


39323 


41545 


43854 


1 


0.35142 


0.37215 


0.39359 


0.41582 


0.43893 





24° 


23° 
156° 


22° 


21° 


20° 


Cot 


155° 


157° 


158° 


159° 





217 



III. LOGARITHMIC 





109° 


108° 


107° 


106° 


105° 


Tan 


70° 


71° 


72° 


73° 


74° 


0' 


0.43893 


0.46303 


0.48822 


0.51466 


0.54250 


1 


43933 


46344 


48865 


51511 


54298 


2 


43972 


46385 


48908 


51557 


54346 


3 


44011 


46426 


48952 


51602 


54394 


4 


44051 


46467 


48995 


51647 


54441 


5 


0.44090 


0.46508 


0.49038 


0.51693 


0.54489 


6 


44130 


46550 


49081 


51738 


54537 


7 


44169 


46591 


49124 


51783 


54585 


8 


44209 


46632 


49167 


51829 


54633 


9 


44248 


46673 


49211 


51874 


54681 


10 


0.44288 


0.46715 


0.49254 


0.51920 


0.54729 


11 


44327 


46756 


49297 


51965 


54778 


12 


44367 


46798 


49341 


52011 


54826 


13 


44407 


46839 


49384 


52057 


54874 


14 


44446 


46880 


49428 


52103 


54922 


15 


0.44486 


0.46922 


0.49471 


0.52148 


0.54971 


16 


44526 


46963 


49515 


52194 


55019 


17 


44566 


47005 


49558 


52240 


55067 


18 


44605 


47047 


49602 


52286 


55116 


19 


44645 


47088 


49645 


52332 


55164 


20 


0.44685 


0.47130 


0.49689 


0.52378 


0.55213 


21 


44725 


47171 


49733 


52424 


55262 


22 


44765 


47213 


49777 


52470 


55310 


23 


44805 


47255 


49820 


52516 


55359 


24 


44845 


47297 


49864 


52562 


55408 


25 


0.44885 


0.47339 


0.49908 


0.52608 


0.55456 


26 


44925 


47380 


49952 


52654 


55505 


27 


44965 


47422 


49996 


52701 


55554 


28 


45005 


47464 


50040 


52747 


55603 


29 


45045 


47506 


500S4 


52793 


55652 


30 


0.45085 


0.47548 


0.50128 


0.52840 


0.55701 


31 


45125 


47590 


50172 


52886 


55750 


32 


45165 


47632 


50216 


52932 


55799 


33 


45206 


47674 


50260 


52979 


55849 


34 


45246 


47716 


50304 


53025 


55898 


35 


0.45286 


0.47758 


0.50348 


0.53072 


0.55947 


36 


45327 


47800 


50393 


53119 


55996 


37 


45367 


47843 


50437 


53165 


56046 


38 


45407 


47885 


50481 


53212 


56095 


39 


45448 


47927 


5052G 


53259 


56145 


40 


0.45488 


0.47969 


0.50570 


0.53306 


0.56194 • 


41 


45529 


48012 


50615 


53352 


56244 


42 


45569 


48054 


50659 


53399 


56293 


43 


45610 


48097 


50704 


53446 


56343 


44 


45650 


48139 


50748 


53493 


56393 


45 


0.45691 


0.48181 


0.50793 


0.53540 


0.56442 


46 


45731 


48224 


50837 


53587 


56492 


47 


45772 


48266 


50882 


53634 


56542 


48 


45813 


48309 


50927 


53681 


56592 


49 


45853 


48352 


50971 


53729 


56642 


50 


0.45894 


0.48394 


0.51016 


0.53776 


0.56692 


51 


45935 


48437 


51061 


53823 


56742 


52 


45975 


48480 


51106 


53870 


56792 


53 


46016 


48522 


51151 


53918 


56842 


54 


46057 


48565 


51196 


53965 


56892 


55 


0.46098 


0.48608 


0.51241 


0.54013 


0.56943 


56 


46139 


48651 


51286 


54060 


56993 


57 


46180 


48694 


51331 


54108 


57043 


58 


46221 


48736 


51376 


54155 


57094 


59 


46262 


48779 


51421 


54203 


57144 


60 


0.46303 


0.48822 


0.51466 


0.54250 


0.57195 




19° 


18° 


17° 


16° 


15° 


Cot 


160° 


161° 


162° 


163° 


164° 



218 



TANGENTS AND COTANGENTS 



104° 


103° 


102° 


101° 


100° 


Tan 


75° 


76° 


77° 
0.63664 


78° 


79° 




0.57195 


0.60323 


0.67253 


0.71135 


60' 


57245 


60377 


63721 


67315 


71202 


59 


57296 


60431 


63779 


67377 


71270 


58 


57347 


60485 


63837 


67439 


71338 


57 


57397 


60539 


63895 


67502 


71405 


56 


0.57448 


0.60593 


0.63953 


0.67564 


0.71473 


55 


57499 


60647 


64011 


67627 


71541 


54 


57550 


60701 


64069 


67689 


71609 


53 


57601 


60755 


64127 


67752 


71677 


53 


57652 


60810 


64185 


67815 


71746 


51 


0.57703 


0.60864 


0.64243 


0.67878 


0.71814 


50 


57754 


60918 


64302 


67941 


71883 


49 


57805 


60973 


64360 


68004 


71951 


48 


57856 


61028 


64419 


68067 


72020 


47 


57907 


61082 


64477 


68130 


720S9 


46 


0.57959 


0.61137 


0.64536 


0.68194 


0.72158 


45 


58010 


61192 


64595 


68257 


72227 


44 


58061 


61246 


64653 


68321 


72296 


43 


58113 


61301 


64712 


68384 


72365 


42 


58164 


61356 


64771 


68448 


72434 


41 


0.58216 


0.61411 


0.64830 


0.68511 


0.72504 


40 


58267 


61466 


64889 


68575 


72573 


39 


58319 


61521 


64949 


68639 


72643 


38 


58371 


61577 


65008 


68703 


72712 


37 


58422 


61632 


65067 


68767 


72782 


36 


0.58474 


0.61687 


0.65126 


0.68832 


0.72852 


35 


58526 


61743 


65186 


68896 


72922 


34 


58578 


61798 


65245 


6S960 


72992 


33 


58630 


61853 


65305 


69025 


73063 


33 


58682 


61909 


65365 


69089 


73133 


31 


0.58734 


0.61965 


0.65424 


0.69154 


0.73203 


30 


58786 


62020 


65484 


69218 


73274 


39 


58839 


62076 


65544 


69283 


73345 


38 


58891 


62132 


65604 


69348 


73415 


37 


58943 


62188 


65664 


69413 


73486 


36 


0.58995 


0.62244 


0.65724 


0.69478 


0.73557 


35 


59048 


62300 


65785 


69543 


73628 


34 


59100 


62356 


65845 


69609 


73699 


33 


59153 


62412 


65905 


69674 


73771 


33 


59205 


62468 


65966 


69739 


73842 


31 


0.59258 


0.62524 


0.66026 


0.69805 


0.73914 


30 


59311 


62581 


66087 


69870 


73985 


19 


59364 


62637 


66147 


69936 


74057 


18 


59416 


62694 


66208 


70002 


74129 


17 


59469 


62750 


66269 


70068 


74201 


16 


0.59522 


0.62807 


0.66330 


0.70134 


0.74273 


15 


59575 


62863 


66391 


70200 


74345 


14 


59628 


62920 


66452 


70266 


74418 


13 


59681 


62977 


66513 


70332 


74490 


13 


59734 


63034 


66574 


70399 


74563 


11 


0.59788 


0.63091 


0.66635 


0.70465 


0.74635 


10 


59841 


63148 


66697 


70532 


74708 


9 


59894 


63205 


66758 


70598 


74781 


8 


59948 


63262 


66820 


70665 


74854 


7 


60001 


63319 


66881 


70732 


74927 


6 


0.60055 


0.63376 


0.66943 


0.70799 


0.75000 


5 


60108 


63434 


67005 


70866 


75074 


4 


60162 


63491 


67067 


70933 


75147 


3 


60215 


63548 


67128 


71000 


75221 


3 


60269 


63606 


67190 


71067 


75294 


1 


0.60323 


0.63664 


0.67253 


0.71135 


0.75368 





14° 


13° 


12° 


11° 


10° 


Cot 


165° 


166° 


167° 


168° 


169° 





219 



III. LOGARITHMIC 





99° 


98° 


97° 


96° 


95° 


Tan 


80° 


81° 


v83° 


83° 


84° 


0' 


i 0.75368 


0.80029 


0.85220 


0.91086 


0.97838 


1 


75442 


80111 


85312 


91190 


97960 


2 


75-516 


80193 


85403 


91295 


98082 


3 


75590 


80275 


85496 


91400 


98204 


4 


75665 


80357 


85588 


91505 


98327 


5 


0.75739 


0.80439 


0.85680 


0.91611 


0.98450 


6 


75814 


80522 


85773 


91717 


98573 


7 


75888 


80605 


85866 


91823 


98697 


8 


75963 


80688 


85959 


91929 


98821 


9 


76038 


80771 


86052 


92036 


98945 


10 


0.76113 


0.80854 


0.86146 


0.92142 


0.99070 


11 


76188 


80937 


86239 


92249 


99195 


12 


76263 


81021 


86333 


92357 


99321 


13 


76339 


81104 


86427 


92464 


99447 


14 


76414 


81188 


86522 


92572 


99573 


15 


0.76490 


0.81272 


0.86616 


0.92680 


0.99699 


16 


76565 


81356 


86711 


92789 


99826 


17 


76641 


81440 


86806 


92897 


99954 


18 


76717 


81525 


86901 


93006 


1.00081 


19 


76794 


81609 


86996 


93115 


00209 


30 


0.76870 


0.81694 


0.87091 


0.93225 


1.00338 


31 


76946 


81779 


87187 


93334 


00466 


33 


77023 


81864 


87283 


93444 


00595 


33 


77099 


81949 


87379 


93555 


00725 


34 


77176 


82035 


87475 


93665 


00855 


35 


0.77253 


0.82120 


0.87572 


0.93776 


1.00985 


36 


77330 


82206 


87668 


93887 


01116 


37 


77407 


82292 


87765 


93998 


01247 


38 


77484 


82378 


87862 


94110 


01378 


39 


77562 


82464 


87960 


94222 


01510 


30 


0.77639 


0.82550 


0.88057 


0.94334 


1.01642 


31 


77717 


82637 


88155 


94447 


01775 


33 


77795 


82723 


88253 


94559 


01908 


33 


77873 


82810 


88351 


94672 


02041 


34 


77951 


82897 


88449 


94786 


02175 


35 


0.78029 


0.82984 


0.88548 


0.94899 


1.02309 


36 


78107 


83072 


88647 


95013 


02444 


37 


78186 


83159 


88746 


95127 


02579 


38 


78264 


83247 


88845 


95242 


02715 


39 


78343 


83335 


88944 


95357 


02850 


40 


0.78422 


0.83423 


0.89044 


0.95472 


1.02987 


41 


78501 


83511 


89144 


95587 


03123 


43 


78580 


83599 


89244 


95703 


03261 


43 


78659 


83688 


89344 


95819 


03398 


44 


78739 


83776 


89445 


95935 


03536 


45 


0.78818 


0.83865 


0.89546 


0.96052 


1.03675 


46 


78898 


83954 


89647 


96168 


03813 


47 


78978 


84044 


89748 


96286 


03953 


48 


79058 


84133 


89850 


96403 


04092 


49 


79138 


84223 


89951 


96521 


04233 


60 


0.79218 


0.84312 


0,90053 


0.96639 


1.04373 


51 


79299 


84402 


90155 


96758 


04514 


63 


79379 


84492 


90258 


95876 


04656 


53 


79460 


84583 


90360 


96995 


04798 


64 


79541 


84673 


90463 


97115 


04940 


65 


0.79622 


0.84764 


0.90566 


0.97234 


1.05083 


66 


79703 


84855 


90670 


97355 


05227 


67 


79784 


84946 


90773 


97475 


05370 


68 


79866 


85037 


90877 


97596 


05515 


69 


79947 


85128 


90981 


97717 


05660 


60 


0.80029 


0.85220 


0.91086 


0.97838 


1.05805 




9° 


8° 


7° 


6° 


5° 


Cot 


170° 


171° 


172° 


153° 


174° J 



220 



TANGENTS AND COTANGENTS 



94° 


93° 


92° 


91° 


90° 


Tan 


85° 


86° 


87° 


88° 


89° 




1.05805 


1.15536 


1.28060 


1.45692 


1.75808 


60' 


05951 


15718 


28303 


46055 


76538 


59 


06097 


15900 


28547 


46422 


77280 


58 


06244 


16084 


28792 


46792 


78036 


57 


06391 


16268 


29038 


47165 


78805 


56 


1.06538 


1.16453 


1.29286 


1.47541 


1.79587 


55 


06687 


16639 


29535 


47921 


80384 


54 


06835 


16825 


29786 


48304 


81196 


53 


06984 


17013 


30038 


48690 


82024 


52 


07134 


17201 


30292 


49080 


82867 


51 


1.07284 


1.17390 


1.30547 


1.49473 


1.83727 


50 


07435 


17580 


30804 


49870 


84605 


49 


07586 


17770 


31062 


50271 


85500 


48 


07738 


17962 


31322 


50675 


86415 


47 


07890 


18154 


31583 


51083 


87349 


46 


1.08043 


1.18347 


1.31846 


1.51495 


1.88304 


45 


08197 


18541 


32110 


51911 


89280 


44 


08350 


18736 


32376 


62331 


90278 


43 


08505 


18932 


32644 


52755 


91300 


43 


08660 


19128 


32913 


53183 


92347 


41 


1.08815 


1.19326 


1.33184 


1.53615 


1.93419 


40 


08971 


19524 


33457 


54052 


94519 


39 


09128 


29723 


33731 


54493 


95647 


38 


09285 


19924 


34007 


54939 


96806 


37 


09443 


20125 


34285 


55389 


97996 


36 


1.09601 


1.20327 


1.34565 


1.55844 


1.99219 


35 


09760 


20530 


34846 


56304 


2.00478 


34 


09920 


20734 


35130 


56768 


01175 


33 


10080 


20939 


35415 


57238 


03111 


33 


10240 


21145 


35702 


57713 


04490 


31 


1.10402 


1.21351 


1.35991 


1.58193 


2.05914 


30 


10563 


21559 


36282 


58679 


07387 


29 


10726 


21768 


36574 


59170 


08911 


28 


10889 


21978 


36869 


59666 


10490 




11052 


22189 


37166 


60168 


12129 


26 


1.11217 


1.22400 


1.37465 


1.60677 


2. 13833 


25 


11382 


22613 


37766 


61191 


15606 


24 


11547 


22827 


38069 


61711 


17454 


23 


11713 


23042 


38374 


62238 


19385 


22 


11880 


23258 


38681 


62771 


21405 


21 


1.12047 


1.23475 


1.38991 


1.63311 


2.23524 


20 


12215 


23694 


39302 


63857 


25752 


19 


12384 


23913 


39616 


64410 


28100 


18 


12553 


24133 


39932 


64971 


30582 


17 


12723 


24355 


40251 


65539 


33215 


16 


1.12894 


1.24577 


1.40572 


1.66114 


2.36018 


15 


13065 


24801 


40895 


66698 


39014 


14 


13237 


25026 


41221 


67289 


42233 


13 


13409 


25252 


41549 


67888 


45709 


12 


13^83 


25479 


41879 


68495 


49488 


11 


1.13757 


1.25708 


1.42212 


1.69112 


2.53627 


10 


13931 


25937 


42548 


69737 


58203 


9 


14107 


26168 


42886 


70371 


63318 


8 


14283 


26400 


43227 


71014 


89118 


7 


14460 


26634 


43571 


71668 


75812 


6 


1.14637 


1.26868 


1.43917 


1.72331 


2.83730 


5 


14815 


27104 


44266 


73004 


2.93421 


4 


14994 


27341 


44618 


73688 


3.05915 


3 


15174 


27580 


44973 


74384 


3.23524 


3 


15354 


27819 


45331 


75090 


3.53627 


1 


1.15536 

4° 


1.28060 


1.45692 


1.75808 


00 





3° 


2° 


1° 


0° 


Cot 


175° 


176° 1 


177° 


178° 


179° 





221 



IV. NATURAL SINES 





179° 


178° 


177° 


176° 


176° 


Sin 


0° 


1° 


2° 
.03490 


3° 


4° 


(y 


.00000 


.01745 


.05234 


.06970 


1 


029 


774 


519 


263 


,07005 


2 


058 


803 


548 


292 


034 


3 


087 


832 


677 


321 


063 


4 


116 


862 


606 


350 


092 


5 


.00145 


.01891 


.03635 


.05379 


.07121 


6 


175 


920 


664 


408 


150 


7 


204 


949 


693 


437 


179 


8 


233 


978 


723 


466 


.• 208 


9 


262 


.02007 


752 


495 


237 


10 


.00291 


.02036 


.03781 


.05524 


.07266 


11 


320 


065 


810 


553 


295 


12 


349 


094 




582 


324 


13 


378 


123 


868 


611 


353 


14 


407 


152 


897 


640 


382 


15 


.00436 


.02181 


.03926 


.05669 


.07411 


16 


465 


211 


955 


698 


440 


17 


495 


240 


984 


727 


469 


18 


524 


269 


.04013 


756 


498 


19 


553 


298 


042 


785 


527 


20 


.00582 


.02327 


.04071 


.05814 


.07556 


21 


611 


356 


100 


844 


685 


22 


640 


385 


129 


873 


614 


23 


669 


414 


159 


902 


643 


24 


698 


443 


188 


931 


672 


25 


.00727 


.02472 


.04217 


.05960 


.07701 


26 


756 


501 


246 


989 


730 


27 


785 


530' 


275 


.06018 


759 


28 


814 


560 


304 


047 


78S 


29 


844 


589 


333 


076 


817 


30 


.00873 


.02618 


.04362 


.06105 


.07846 


31 


902 


647 


391 


134 


876 


32 


931 


676 


420 


163 


904 


33 


960 


705 


449 


192 


933 


34 


989 


734 


478 


221 


962 


35 


.01018 


.02763 


.04507 


.06250 


.07991 


36 


047 


792 


636 


279 


.08020 


37 


076 


821 


565 


308 


049 


38 


105 


850 


594 


337 


078 


39 


134 


879 


623 


366 


107 


40 


.01164 


.02908 


.04653 


.06395 


.08136 


41 


193 


938 


682 


424 


165 


42 


222 


967 


711 


453 


194 


43 


251 


996 


740 


482 


223 


44 


280 


.03025 


769 


511 


252 


45 


„01309 


.03054 


,04798 


.06540 


.08281 


46 


338 


083 


827 


569 


310 


47 


367 


112 


856 


698 


339 


48 


396 


141 


885 


627 


368 


49 


425 


170 


914 


656 


397 


50 


.01454 


.03199 


.04943 


.06685 


.08426 


51 


483 


228 


972 


714 


455 


52 


513 


257 


.05001 


743 


484 


53 


542 


286 


030 


773 


513 


54 


671 


316 


059 


802 


642 


55 


.01600 


.03345 


.05088 


.06831 


.08571 


66 


629 


374 


117 


860 


600 


57 


658 


403 


146 


889 


629 


58 


687 


432 


175 


918 


658 


59 


716 


461 


205 


947 


687 


60 


.01745 


.03490 


.05234 


.06976 


.08716 




89° 


88° 


87° 


86° 


85** 


Cos 


90° 


91° 


92° 


93° 


940 



222 



AND COSINES. 



f 174° 


1 173° 


172° 


171° 


170° 
9° 


$in 


5° 


6° 


7° 


8° 




.08716 


.10453 


.12187 


.13917 


.15643 


60' 


743 


482 


216 


946 


672 


59 


774 


511 


245 


975 


701 


58 


803 


540 


274 


. 14004 


730 


57 


831 


569 


302 


033 


758 


6G 


.08860 


.10597 


.12331 


.14061 


.15787 


55 


889 


626 


360 


090 


816 


54 


918 


655 


389 


119 


845 


53 


947 


684 


418 


148 


873 


62 


976 


713 


447 


177 


902 


51 


.09005 


.10742 


.12476 


.14205 


.15931 


50 


034 


771 


504 


234 


959 


49 


063 


800 


533 


263 


988 


48 


092 


829 


562 


292 


.16017 


47 


121 


858 


591 


320 


046 


46 


.09150 


.10887 


.12620 


.14349 


.16074 


45 


179 


916 


649 


378 


103 


44 


208 


945 


678 


407 


132 


43 


237 


973 


706 


436 


160 


42 


266 


11002 


735 


464 


189 


41 


.09295 


.11031 


.12764 


.14493 


.16218 


40 


324 


060 


793 


522 


246 


39 


353 


089 


822 


551 


275 


38 


382 


118 


851 


580 


304 


37 


411 


147 


880 


608 


333 


36 


.09440 


.11176 


.12908 


.14637 


.16361 


35 


469 


205 


937 


666 


390 


34 


498 


234 


966 


695 


419 


33 


527 


263 


995 


723 


447 


32 


556 


291 


. 13024 


752 


476 


31 


.09585 


.11320 


.13053 


.14781 


.16505 


30 


614 


349 


081 


810 


533 


29 


642 


378 


110 


838 


562 


28 


671 


407 


139 


867 


591 


27 


700 


436 


168 


890 


620 


26 


.09729 


.11465 


.13197 


.14925 


.16648 


25 


758 


494 


226 


954 


677 


24 


787 


523 


254 


982 


706 


23 


816 


552 


283 


.15011 


734 


22 


845 


580 


312 


040 


763 


21 


.09874 


.11609 


.13341 


.15069 


.16792 


20 


903 


638 


370 


097 


820 


19 


932 


667 


399 


126 


849 


18 


961 


096 


427 


155 


878 


17 


990 


725 


456 


184 


906 


16 


.10019 


.11754 


.13485 


.15212 


.16935 


15 


048 


783 


514 


241 


964 


14 


077 


812 


543 


270 


992 


13 


106 


840 


572 


299 


.17021 


12 


135 


869 


600 


327 


050 


11 


.10164 


.11898 


.13629 


.15356 


.17078 


10 


192 


927 


658 


385 


107 


9 


221 


956 


687 


414 


136 


8 


250 


985 


716 


442 


164 


7 


279 


.12014 


744 


471 


193 


6 


.10308 


.12043 


.13773 


.15500 


.17222 


5 


337 


071 


802 


52J 


250 


4 


366 


100 


831 


557 


279 


3 


395 


129 


860 


586 


308 


2 


424 


158 


889 


615 


336 


1 


.10453 


.12187 


.13917 


.15643 


.17365 





84° 
95° 


83° 


82° 


81° 1 


80° 


Cos 


96° 


97° 


98° 1 


99° 





223 



IV. NATURAL SINES 





169° 


168° 


167° 


166° 


165° 
14° 


Sin 


10° 


11° 


13° 


13° 


0' 


.17365 


.19081 


.20791 


.22495 


.24192 


1 


393 


109 


820 


523 


220 


2 


422 


138 


848 


552 


249 


3 


451 


167 


877 


580 


277 


4 


479 


195 


905 


608 


305 


5 


.17508 


.19224 


.20933 


.22637 


,24333 


6 


537 


252 


962 


665 


362 


7 


565 


281 


990 


693 


390 


8 


594 


309 


.21019 


722 


418 


9 


623 


338 


047 


750 


446 


10 


.17651 


.19366 


.21076 


.22778 


.24474 


11 


680 


395 


104 


807 


503 


13 


708 


423 


132 


835 


531 


13 


737 


452 


161 


803 


559 


14 


766 


481 


189 


892 


587 


15 


.17794 


.19509 


.21218 


.22920 


.24615 


16 


823 


538 


246 


948 


644 


17 


852 


566 


275 


977 


672 


18 


880 


595 


303 


.23005 


700 


19 


909 


623 


331 


033 


728 


20 


.17937 


.19652 


.21360 


.23062 


.24756 


31 


966 


680 


388 


090 


784 


23 


995 


709 


417 


118 


813 


23 


.18023 


737 


445 


146 


841 


24 


052 


766 


474 


175 


869 


25 


.18081 


.19794 


.21502 


.23203 


.24897 


26 


109 


823 


530 


231 


925 


27 


138 


851 


559 


260 


954 


28 


166 


880 


587 


288 


982 


29 


195 


908 


616 


316 


. 25010 


30 


.18224 


.19937 


.21644 


.23345 


.25038 


31 


252 


965 


672 


373 


066 


32 


281 


994 


701 


401 


094 


33 


309 


.20022 


729 


429 


122 


34 


338 


051 


758 


458 


151 


35 


.18367 


.20079 


.21786 


.23486 


.25179 


36 


395 


108 


814 


514 


207 


37 


424 


136 


843 


542 


235 


38 


452 


165 


871 


571 


263 


39 


481 


193 


899 


599 


291 


40 


.18509 


.20222 


.21928 


.23627 


.25320 


41 


538 


250 


956 


656 


348 


42 


567 


279 


985 


684 


376 


43 


595 


307 


.22013 


712 


404 


44 


624 


336 


041 


740 


432 


45 


.18652 


.20364 


.22070 


.23769 


.25460 


46 


681 


393 


098 


797 


488 


47 


710 


421 


126 


825 


516 


48 


738 


450 


155 


853 


545 


49 


767 


478 


183 


882 


573 


50 


.18795 


.20507 


.22212 


.23910 


.25601 


51 


824 


535 


240 


938 


629 


53 


852 


563 


268 


966 


657 


53 


881 


592 


297 


995 


685 


54 


910 


620 


325 


.24023 


713 


55 


.18938 


.20649 


.22353 


.24051 


.25741 


56 


967 


677 


382 


079 


709 


57 


995 


700 


410 


108 


798 


58 


.19024 


734 


438 


136 


826 


59 


052 


763 


407 


164 


854 


60 


.19081 


.20791 


.22495 


.24192 


-.25882 




79° 


78° 


77° 


76° 


75° 


Cos 


100° 


101° 


102° 


103° 


104° 



224 



AND COSINES. 










164° 


163° 


162° 


161° 


160° 


Sin 


15° 


16° 


17° 


18° 


19° 




.25882 


.27564 


.29237 


.30902 


.32557 


60' 


910 


592 


265 


929 


584 


59 


938 


620 


293 


957 


612 


58 


966 


648 


321 


985 


639 


57 


994 


676 


348 


.31012 


667 


56 


, .26022 


.27704 


.29376 


.31040 


.32694 


55 


1 050 


731 


404 


068 


722 


54 


079 


759 


432 


095 


749 


53 


107 


787 


460 


123 


777 


53 


135 


815 


487 


151 


804 


51 


, .26163 


.27843 


.29515 


.31178 


.32832 


50 


! 191 


871 


543 


206 


859 


49 


219 


899 


571 


233 


887 


48 


! 247 


927 


599 


261 


914 


47 


1 275 


955 


626 


289 


942 


46 


.26303 


.27983 


.29654 


.31316 


.32969 


45 


331 


.28011 


682 


344 


997 


44 


359 


039 


710 


372 


.33024 


43 


387 


067 


737 


399 


051 


43 


415 


095 


765 


427 


079 


41 


' .26443 


.28123 


.29793 


.31454 


.33106 


40 


471 


150 


821 


482 


134 


39 


500 


178 


849 


510 


161 


38 


528 


206 


876 


537 


189 


37 


556 


234 


904 


565 


216 


36 


.26584 


.28262 


.29932 


.31593 


.33244 


35 


1 612 


290 


960 


620 


271 


34 


1 640 


318 


987 


648 


298 


33 


668 


346 


.30015 


675 


326 


33 


696 


374 


043 


703 


353 


31 


.26724 


.28402 


.30071 


.31730 


.33381 


30 


1 752 


429 


098 


758 


408 


29 


780 


457 


126 


786 


436 


38 


808 


485 


154 


813 


463 


37 


836 


513 


182 


841 


490 


26 


.26864 


.28541 


.30209 


.31868 


.33518 


35 


892 


569 


237 


896 


545 


34 


920 


597 


265 


923 


573 


33 


948 


625 


292 


951 


600 


33 


976 


652 


320 


979 


627 


31 


.27004 


.28680 


.30348 


.32006 


.33655 


30 


1 032 


708 


376 


034 


682 


19 


060 


736 


403 


061 


710 


18 


088 


764 


431 


089 


737 


17 


116 


792 


459 


116 


764 


16 


.27144 


.28820 


.30486 


.32144 


.33792 


15 


172 


847 


514 


171 


819 


14 


200 


875 


542 


199 


846 


13 


228 


903 


570 


227 


874 


13 


256 


931 


597 


254 


901 


11 


.27284 


.28959 


.30625 


.32282 


.38929 


10 


312 


987 


653 


309 


956 


9 


340 


.29015 


680 


337 


983 


8 


368 


042 


708 


364 


.34011 


7 


396 


070 


736 


392 


038 


6 


.27424 


.29098 


.30763 


.32419 


.34065 


5 


452 


126 


791 


447 


093 


4 


4S0 


154 


819 


474 


120 


3 


508 


182 


846 


502 


147 


3 


536 


209 


874 


529 


175 


1 


.27564 


.29237 


.30902 


.32557 


.34202 





74° 


73° 


72° 


71° 


70° 


Cos 


.105° 


106° 


107° 


108° 


109° 




225 



IV. NATURAL SINES 





159° 


158° 


157° 


156° 


155° 


Sin 


30° 


31° 


32° 


33° 


34° 


0' 


.34202 


.35837 


.37461 


.39073 


.40674 


1 


229 


864 


488 


100 


700 


2 


257 


891 


515 


127 


727 1 


3 


284 


918 


542 


153 


753 ( 


4 


311 


945 


569 


180 


780 


6 


.34339 


.35973 


.37595 


.39207 


.40806 1 


6 


366 


.36000 


622 


234 


833 1 


7 


393 


027 


649 


260 


860 


8 


421 


054 


676 


287 


886 


9 


448 


081 


703 


314 


913 


10 


.34475 


.36108 


.37730 


.39341 


.40939 


11 


503 


135 


757 


367 


966 


13 


530 


162 


784 


394 


992 


13 


557 


190 


811 


421 


.41019 


14 


584 


217 


838 


448 


045 1 


15 


.34612 


.36244 


.37865 


.39474 


.41072 


16 


639 


271 


892 


501 


098 


17 


666 


298 


919 


528 


125 1 


18 


694 


325 


946 


555 


151 


19 


721 


352 


973 


581 


178 ; 


30 


.34748 


.36379 


.37999 


.39608 


.41204 


31 


775 


406 


.38026 


635 


231 


33 


803 


434 


053 


661 


257 1 


33 


830 


461 


080 


688 


284 


34 


857 


488 


107 


715 


310 


35 


.34884 


.36515 


.38134 


.39741 


.41337 


36 


912 


542 


161 


768 


363 


37 


939 


569 


188 


795 


390 1 


38 


966 


596 


215 


822 


416 


39 


993 


623 


241 


848 


443 


30 


.35021 


.36650 


.38268 


.39875 


.41469 


31 


048 


677 


295 


902 


496 ) 


33 


075 


704 


322 


928 


522 


33 


102 


731 


349 


955 


549 


34 


130 


758 


376 


982 


575 


35 


.35157 


.36785 


.38403 


.40008 


.41602 


36 


184 


812 


430 


035 


628 


37 


211 


839 


456 


062 


655 


38 


239 


867 


483 


088 


681 


39 


266 


894 


510 


115 


707 


40 


.35293 


.36921 


.38537 


.40141 


.41734 


41 


320 


948 


564 


168 


760 


43 


347 


975 


591 


195 


787 


43 


375 


.37002 


617 


221 


813 


44 


402 


029 


644 


248 


840 


45 


.35429 


.37056 


.38671 


.40275 


.41866 


46 


450 


083 


698 


301 


892 


47 


484 


110 


725 


328 


919 


48 


511 


137 


752 


355 


945 


49 


538 


164 


778 


381 


972 


50 


.35565 


.37191 


.38805 


.40408 


.41998 


51 


592 


218 


832 


434 


.42024 


53 


619 


245 


859 


461 


051 


53 


647 


272 


886 


488 


077 


54 


674 


299 


912 


514 


104 


55 


.35701 


.37326 


.38939 


.40541 


.42130 


56 


728 


353 


966 


567 


156 


57 


755 


380 


993 


594 


183 


58 


782 


407 


.39020 


621 


209 


59 


810 


434 


046 


647 


235 


60 


.35837 


.37461 


.39073 


.40674 


.42262 




69° 


68° 


67° 


66° 


65° 


Cos 


110° 


111° 


112° 


113° 


114° 






21 


26 






^^^^■I^^^^H 


I^^^^H 






^^■I^H 





AND COSINES 



^^ 



154° 


153° 


152° 


151° 


150° 


Sin 


35° 


36° 


37° 


38° 


39° 




.42262 


.43837 


.45399 


.46947 


.48481 


6(K 


288 


863 


425 


973 


506 


59 


315 


. 889 


451 


999 


532 


58 


341 


916 


477 


47024 


557 


57 


367 


942 


503 


050 


583 


56 


.42394 


.43968 


.45529 


.47076 


.48608 


55 


420 


994 


554 


101 


634 


54 


446 


.44020 


580 


127 


659 


53 


473 


046 


606 


153 


684 


53 


499 


072 


632 


178 


710 


51 


.42525 


.44098 


.45658 


.47204 


.48735 


50 


552 


124 


684 


229 


761 


49 


578 


151 


710 


255 


786 


48 


604 


177 


736 


281 


811 


47 


631 


203 


762 


306 


837 


46 


.42657 


.44229 


.45787 


.47332 


.48862 


45 


683 


255 


813 


358 


888 


44 


709 


281 


839 


383 


913 


43 


736 


307 


865 


409 


938 


43 


762 


333 


891 


434 


964 


41 


.42788 


.44359 


.45917 


.47460 


.48989 


40 


815 


385 


942 


486 


.49014 


39 


841 


411 


968 


511 


040 


38 


867 


437 


994 


537 


065 


37 


894 


464 


.46020 


562 


090 


36 


.42920 


.44490 


.46046 


.47588 


.49116 


35 


946 


516 


072 


614 


141 


34 


972 


542 


097 


639 


166 


33 


. 999 


568 


123 


665 


192 


33 


.43025 


594 


149 


690 


217 


31 


.43051 


.44620 


.46175 


.47716 


.49242 


30 


077 


646 


201 


741 


268 


39 


104 


672 


226 


767 


293 


38 


130 


698 


252 


793 


318 


37 


156 


724 


278 


818 


344 


36 


.43182 


.44750 


.46304 


.47844 


.49369 


35 


209 


776 


330 


869 


394 


34 


235 


802 


355 


895 


419 


33 


261 


828 


381 


920 


445 


33 


287 


854 


407 


946 


470 


31 


.43313 


.44880 


.46433 


.47971 


.49495 


30 


340 


906 


458 


997 


521 


19 


3G6 


932 


484 


.48022 


546 


IS 


392 


958 


510 


048 


571 


17 


418 


984 


536 


073 


596 


16 


.43445 


.45010 


.46561 


.48099 


.49622 


15 


471 


036 


587 


124 


647 


14 


497 


062 


613 


150 


672 


13 


523 


088 


639 


175 


697 


13 


549 


114 


664 


201 


723 


11 


.43575 


.45140 


.46690 


.48226 


.49748 


10 


602 


165 


716 


252 


773 


9 


628 


192 


742 


277 


798 


8 


654 


218 


767 


303 


824 


7 


680 


243 


793 


328 


849 


6 


.43706 


.45269 


.46819 


.48354 


.49874 


5 


733 


295 


844 


379 


899 


4 


759 


321 


870 


405 


924 


3 


785 


347 


. 896 


430 


950 


3 


811 


373 


921 


456 


975 


1 


.43837 


.45399 


.46947 


.48481 


.50000 





64° 


63° 


63° 


61° 


60° 


Cos 


, 115° 


116° 


117° 


118° 


119° 





221 



IV. NATURAL SINES 





149° 


148° 


147° 


146° 


145° 


Sin 


30° 


31° 


33° 


33° 


34° 


0' 


.50000 


.51504 


.52992 


.54464 


.55919 


1 


025 


529 


.53017 


488 


943 


3 


050 


554 


041 


513 


968 


3 


076 


579 


066 


537 


992 


4 


101 


604 


091 


561 


.56016 


5 


.50126 


.51628 


.53115 


.54586 


.56040 


6 


151 


653 


140 


610 


064 


7 


176 


678 


164 


635 


088 


8 


201 


703 


189 


659 


112 


9 


227 


728 


214 


683 


136 


10 


.50252 


.51753 


.53238 


.54708 


.56160 


11 


277 


778 


263 


732 


184 


13 


302 


803 


288 


756 


208 i 


13 


327 


828 


312 


781 


232 ! 


14 


352 


852 


337 


805 


256 ; 


15 


.50377 


.51877 


.53361 


.54829 


.56280 


16 


403 


902 


386 


854 


305 ; 


17 


428 


927 


411 


878 


329 


18 


453 


952 


435 


902 


353 ! 


19 


478 


977 


460 


927 


377 


30 


.50503 


.52002 


.53484 


.54951 


.56401 ' 


31 


528 


026 


509 


975 


425 


33 


553 


051 


534 


999 


449 ; 


33 


578 


076 


558 


.55024 


473 


34 


603 


101 


583 


048 


497 i 


35 


.50628 


.52126 


.53607 


.55072 


.56521 : 


36 


654 


151 


632 


097 


545 i 


37 


679 


175 


656 


121 


569 1 


38 


704 


200 


681 


145 


593 


39 


729 


225 


705 


169 


617 


30 


.50754 


.52250 


.53730 


.55194 


.56641 


31 


779 


275 


754 


218 


605 1 


33 


804 


299 


779 


242 


689 


33 


829 


324 


804 


266 


713 ! 


34 


854 


349 


828 


291 


736 


35 


.50879 


.52374 


.53853 


.55315 


.56760 , 


36 


904 


399 


877 


339 


784 


37 


929 


423 


902 


363 


808 


38 


954 


448 


926 


388 


832 


39 


979 


473 


951 


412 


856 


40 


.51004 


.52498 


.53975 


.55436 


.56880 


41 


029 


522 


.54000 


460 


904 


43 


054 


547 


024 


484 


928 


43 


079 


572 


049 


509 


952 


44 


104 


597 


073 


533 


976 


45 


.51129 


.52621 


.54097 


.55557 


.57000 


46 


154 


646 


122 


581 


024 


47 


179 


671 


146 


605 


047 


48 


204 


696 


171 


630 


071 


49 


229 


720 


195 


654 


095 


50 


.51254 


.52745 


.54220 


.55678 


.57119 


51 


279 


770 


244 


702 


143 


53 


304 


794 


269 


726 


167 


53 


329 


819 


293 


750 


191 


54 


354 


844 


317 


775 


215 


55 


.51379 


.52869 


.54342 


.55799 


.57238 


56 


404 


893 


366 


823 


262 


57 


429 


918 


391 


847 


286 


58 


454 


943 


415 


~ 871 


310 


59 


479 


967 


440 


895 


334 


60 


.51504 


.52992 


.54464 


.55919 


.57358 ' 




59° 

120° 


58° 


67° 


56° 


55° 


Cos 


121° 


122° 


123° 


124° 


1 








■^^^^^^BB 







AND COSINES 





144° 
35° 


143° 


142° 


141° 


140° 


Sin 


36° 


37° 


38° 


39° 




.57358 


.58779 


.60182 


.61566 


.62932 


60' 




381 


802 


205 


589 


955 


59 




405 


826 


228 


612 


977 


58 




429 


849 


251 


635 


.63000 


57 




453 


873 


274 


658 


022 


56 


1 


.57477 


.58896 


.60298 


.61681 


.63045 


55 


I 


501 


920 


321 


704 


068 


54 




524 


943 


344 


726 


090 


53 




548 


967 


367 


749 


113 


53 




572 


990 


390 


772 


135 


51 




.57596 


.59014 


.60414 


.61795 


.63158 


50 




619 


037 


437 


818 


180 


49 




643 


061 


460 


841 


203 


48 




607 


084 


483 


864 


225 


47 




691 


108 


506 


887 


248 


46 


1' 


.57715 


.59131 


.60529 


.61909 


.63271 


45 




738 


154 


553 


932 


293 


44 




702 


178 


576 


955 


316 


43 




786 


201 


599 


978 


338 


43 




810 


225 


622 


.62001 


361 


41 




.57833 


.59248 


.60645 


.62024 


.63383 


40 




857 


272 


668 


046 


406 


39 




881 


295 


691 


069 


428 


38 




904 


318 


714 


092 


451 


37 




928 


342 


738 


115 


473 


36 




.57952 


.59365 


.60761 


.62138 


.63496 


35 




976 


389 


784 


160 


518 


34 




999 


412 


807 


183 


540 


33 




.58023 


436 


830 


206 


563 


33 




047 


459 


853 


229 


585 


31 




.58070 


.59482 


.60876 


.62251 


.63608 


30 




094 


506 


899 


274 


630 


39 




118 


529 


922 


297 


653 


38 




141 


552 


945 


320 


675 


37 




165 


576 


908 


342 


698 


36 




.58189 


.59599 


.60991 


62365 


.63720 


35 




212 


622 


.61015 


388 


742 


34 




236 


646 


038 


411 


765 


33 




200 


669 


061 


433 


787 


33 




283 


693 


084 


456 


810 


31 




.58307 


.59716 


.61107 


.62479 


.63832 


30 




330 


739 


130 


502 


854 


19 




354 


763 


153 


524 


877 


18 




378 


786 


176 


547 


899 


17 




401 


809 


199 


570 


922 


16 


1 


.58425 


.59832 


.61222 


.62592 


.63944 


15 


1 


449 


856 


245 


615 


966 


14 




472 


879 


268 


638 


989 


13 


1 


496 


902 


291 


660 


.64011 


13 




519 


926 


314 


683 


033 


11 




.58543 


.59949 


.61337 


.62706 


.64056 


10 




567 


972 


360 


728 


078 


9 




590 


995 


383 


751 


100 


8 




614 


.60019 


406 


774 


123 


7 




637 


042 


429 


796 


145 


6 




.58661 


.60065 


.61451 


.62819 


.64167 


5 




684 


089 


474 


842 


190 


4 




708 


112 


497 


864 


212 


3 




731 


135 


520 


887 


234 


3 




755 


158 


543 


909 


256 


1 




.58779 


.60182 


.61566 


.62932 


.64279 







54° 


53° 


52° 


51° 


50° 


Cos 




125° 


126° 


127° 


128° 


129° 





229 



IV. NATURAL SINES 





139° 


138° 


137° 


136° 


135° 


Sin 


40° 


41° 


42° 


43° 


44° 


0' 


.64279 


.65606 


.66913 


.68200 


.69466 


1 


301 


628 


935 


221 


487 


2 


323 


650 


756 


242 


508 


3 


346 


672 


978 


264 


529 


4 


368 


694 


999 


285 


549 


5 


.64390 


.65716 


.67021 


.68306 


.69570 


6 


412 


738 


043 


327 


591 


7 


435 


759 


064 


349 


612 


8 


457 


781 


086 


370 


633 


9 


479 


803 


107 


391 


654 


10 


.64501 


.65825 


.67129 


.68412 


.69675 


11 


524 


847 


151 


434 


696 


13 


546 


869 


172 


455 


717 


13 


568 


891 


194 


476 


737 


14 


590 


913 


215 


497 


758 


15 


.64612 


.65935 


.67237 


.68518 


.69779 


16 


635 


956 


258 


539 


800 


17 


657 


978 


280 


561 


821 


18 


679 


.66000 


301 


582 


842 


19 


701 


022 


323 


603 


862 


20 


.64723 


.66044 


.67344 


.68624 


.69883 


21 


746 


066 


366 


645 


904 


22 


768 


088 


387 


666 


925 


23 


790 


109 


409 


688 


946 


24 


812 


131 


430 


709 


966 


25 


.64834 


.66153 


.67452 


.68730 


.69987 


26 


856 


175 


473 


751 


.70008 


27 


878 


197 


495 


772 


029 


28 


901 


218 


516 


793 


049 


29 


923 


240 


538 


814 


070 


30 


.64945 


.66262 


.67559 


.68835 


.70091 


31 


967 


284 


580 


857 


112 


32 


989 


306 


602 


878 


132 


33 


.65011 


327 


623 


899 


153 


34 


033 


349 


645 


920 


174 


35 


.65055 


.66371 


.67666 


.68941 


.70195 


36 


077 


393 


688 


962 


215 


37 


100 


414 


709 


983 


236 


38 


122 


436 


730 


.69004 


257 


39 


144 


458 


752 


025 


277 


40 


.65166 


.66480 


.67773 


.69046 


.70298 


41 


188 


501 


795 


067 


319 


42 


210 


523 


816 


088 


339 


43 


232 


545 


837 


109 


360 


44 


254 


566 


859 


130 


381 


45 


.65276 


.66588 


.67880 


.69151 


.70401 


46 


298 


610 


901 


172 


422 


47 


320 


632 


923 


193 


443 


• 48 


342 


653 


944 


214 


463 


49 


364 


675 


965 


235 


484 


50 


.65386 


.66697 


.67987 


.69256 


.70505 


51 


408 


718 


.68008 


277 


525 


52 


430 


740 


029 


298 


546 


63 


452 


762 


051 


319 


567 


54 


474 


783 


072 


340 


587 


65 


.65496 


.66805 


.68093 


.69361 


.70608 


66 


518 


827 


115 


382 


628 


57 


540 


848 


136 


403 


649 


58 


562 


870 


157 


424 


670 


69 


584 


891 


179 


445 


690 


60 


.65606 


.66913 


.68200 


.69466 


.70711 




49° 


48° 


47° 


46° 


45° 


Cos 


130° 


131° 


132° 


133° 


134° 



230 



AND COSINES 



134° 


133° 
46° 


132° 


131° 


130° 


Sin 


45° 


47° 


48° 


49° 




.70711 


.71934 


.73135 


.74314 


.75471 


ec 


731 


954 


155 


334 


490 


59 


752 


974 


175 


353 


509 


58 


772 


995 


195 


373 


528 


57 


793 


.72015 


215 


392 


547 


56 


.70813 


.72035 


.73234 


.74412 


.75566 


55 


834 


055 


254 


431 


585 


54 


855 


075 


274 


451 


604 


53 


875 


095 


294 


470 


623 


5.3 


896 


116 


314 


489 


642 


51 


.70916 


.72136 


.73333 


.74509 


.75661 


50 


937 


156 


353 


528 


680 


49 


957 


176 


373 


548 


700 


48 


978 


196 


393 


567 


719 


47 


998 


216 


413 


586 


738 


46 


.71019 


.72236 


.73432 


.74606 


.75756 


45 


039 


257 


452 


625 


775 


44 


059 


277 


472 


644 


794 


43 


080 


297 


491 


604 


813 


43 


100 


317 


511 


683 


832 


41 


.71121 


.72337 


.73531 


.74703 


.75851 


40 


141 


357 


551 


722 


870 


39 


162 


377 


570 


741 


889 


38 


182 


397 


590 


760 


908 


37 


203 


417 


610 


780 


927 


36 


.71223 


.72437 


.73629 


.74799 


.75946 


35 


243 


457 


649 


818 


965 


34 


264 


477 


669 


838 


984 


33 


284 


497 


688 


857 


.76003 


32 


305 


517 


708 


876 


022 


31 


.71325 


.72537 


.73728 


.74896 


.76041 


30 


345 


557 


747 


915 


059 


29 


366 


577 


767 


934 


078 


28 


386 


597 


787 


953 


097 


27 


407 


617 


806 


973 


116 


26 


.71427 


.72637 


.73826 


.74992 


.76135 


25 


447 


657 


846 


.75011 


154 


24 


468 


677 


865 


030 


173 




488 


697 


885 


050 


192 


22 


508 


717 


904 


069 


210 


21 


.71529 


.72737 


.73924 


.75088 


.76229 


20 


549 


757 


944 


107 


248 


19 


569 


777 


963 


126 


267 


18 


• 590 


797 


983 


146 


286 


17 


610 


817 


.74002 


165 


304 


16 


.71630 


.72837 


.74022 


.75184 


.76323 


15 


650 


857 


041 


203 


342 


14 


671 


877 


061 


222 


361 


13 


691 


897 


080 


241 


380 


12 


711 


917 


100 


261 


398 


11 


.71732 


.72937 


.74120 


.75280 


.76417 


10 


752 


957 


139 


299 


436 


9 


772 


976 


159 


318 


455 


8 


792 


996 


178 


337 


473 


7 


813 


.73016 


198 


356 


492 


6 


.71833 


.73036 


.74217 


.75375 


.76511 


5 


853 


056 


237 


395 


530 


4 


873 


076 


256 


414 


548 


3 


894 


096 


276 


433 


567 


2 


914 


116 


295 


452 


586 


1 


.71934 


.73135 


.74314 


.75471 


.76604 





44° 


43° 


42° 


41° 


40° 


Cos 


135° 


136° 


137° 


138° 


139° 





231 









IV. NATURAL SINES , 




129° 


128° 


127° 


126° 


125° 


Sin 


50° 


51° 


53° 


53° 


54° 


C 


.76604 


.77715 


.78801 


.79864 


.80902 ' 


1 


623 


733 


819 


881 


919 


2 


642 


• 751 


837 


899 


936 


3 


661 


769 


855 


916 


953 i 


4 


679 


788 


873 


934 


970 


5 


.76698 


.77806 


.78891 


.79951 


.80987 1 


6 


717 


824 


908 


968 


.81004 1 


7 


735 


843 


926 


986 


021 : 


8 


754 


861 


944 


.80003 


038 i 


9 


772 


879 


962 


021 


055 


10 


.76791 


.77897 


.78980 


.80038 


.81072 1 


11 


810 


916 


998 


056 


089 ! 


13 


828 


934 


.79016 


073 


106 j 


13 


847 


952 


033 


091 


123 ! 


14 


866 


970 


051 


108 


140 i 


15 


.76884 


.77988 


.79069 


.80125 


.81157 


16 


903 


.78007 


087 


143 


174 : 


17 


921 


025 


105 


160 


191 


18 


940 


043 


122 


178 


208 


19 


959 


061 


140 


195 


225 


30 


.76977 


.78079 


.79158 


.80212 


.81242 


31 


996 


098 


176 


230 


259 


33 


.77014 


116 


193 


247 


276 i 


33 


033 


134 


211 


264 


293 ! 


34 


051 


152 


229 


282 


310 


35 


.77070 


.78170 


.79247 


.80299 


.81327 


36 


088 


188 


264 


316 


344 


37 


107 


206 


282 


334 


361 


38 


125 


225 


300 


351 


378 


39 


144 


243 


318 


368 


395 1 


30 


.77162 


.78261 


.79335 


.80386 


.81412 1 


31 


181 


279 


353 


403 


428 


33 


199 


297 


371 


420 


445 


33 


218 


315 


388 


438 


462 


34 


236 


333 


400 


455 


479 


35 


.77255 


.78351 


.79424 


.80472 


.81496 


36 


273 


369 


441 


489 


513 


37 


292 


387 


459 


507 


530 


38 


310 


405 


477 


524 


546 


39 


329 


424 


494 


541 


563 


40 


.77347 


.78442 


.79512 


.80558 


.81580 


41 


366 


460 


530 


576 


597 


43 


384 


478 


547 


593 


614 


43 


402 


496 


565 


610 


631 


44 


421 


514 


583 


627 


647 


45 


.77439 


.78532 


.79600 


.80644 


.81664 


46 


458 


550 


618 


662 


681 


47 


476 


568 


635 


679 


698 ( 


48 


494 


586 


653 


696 


714 i 


49 


513 


604 


671 


713 


731 


50 


.77531 


.78622 


.79688 


.80730 


.81748 1 


51 


550 


640 


706 


748 


765 


53 


568 


658 


723 


765 


782 


53 


586 


676 


741 


782 


798 1 


54 


605 


694 


758 


799 


815 ; 


55 


.77623 


.78711 


.79776 


.80816 


.81832 j 


56 


641 


729 


793 


833 


848 ' 


57 


660 


747 


811 


850 


865 


58 


678 


765 


829 


867 


882 


59 


696 


783 


846 


885 


899 , 


60 


.77715 


.78801 


.79864 


.80902 


.81915 , 




39° 


38° 


37° 


36° 


35° 


Cos 


140° 


141° 


142° 


143° 


144° , 


232 1 



AND COSINES 



124° 


123° 


122° 


121° 


120° 


Sin 


55° 


56° 


57° 


58° 


59° 




.81915 


.82904 


.83867 


.84805 


.85717 


60' 


932 


920 


883 


820 


732 


59 


949 


936 


899 


836 


747 


58 


965 


953 


915 


851 


762 


57 


982 


969 


930 


866 


777 


56 


.81999 


.82985 


.83946 


.84882 


.85792 


55 


.82015 


.83001 


962 


897 


806 


54 


032 


017 


978 


913 


821 


53 


048 


034 


994 


928 


836 


52 


065 


050 


.84009 


943 


851 


51 


.82082 


.83066 


.84025 


.84959 


.85866 


50 


098 


082 


041 


974 


881 


49 


' 115 


098 


057 


989 


896 


48 


132 


115 


072 


.85005 


911 


47 


148 


131 


088 


020 


926 


46 


.82165 


.83147 


.84104 


.85035 


.85941 


45 


181 


163 


120 


051 


956 


44 


198 


179 


135 


066 


970 


43 


214 


195 


151 


081 


985 


43 


231 


212 


167 


096 


.86000 


41 


.82248 


.83228 


.84182 


.85112 


.86015 


40 


264 


244 


198 


127 


030 


39 


281 


260 


214 


142 


045 


38 


297 


276 


230 


157 


059 


37 


314 


292 


245 


173 


074 


36 


.82330 


.83308 


.84261 


.85188 


.86089 


35 


347 


324 


277 


203 


104 


34 


363 


340 


292 


218 


119 


33 


: 380 


356 


308 


234 


133 


33 


396 


373 


324 


249 


148 


31 


.82413 


.83389 


.84339 


.85264 


.86163 


30 


429 


405 


355 


279 


178 


29 


446 


421 


370 


294 


192 


28 


462 


437 


386 


310 


207 


27 


478 


453 


402 


325 


222 


26 


.82495 


.83469 


.84417 


.85340 


.86237 


25 


511 


485 


433 


355 


251 


24 


528 


501 


448 


370 


266 


23 


544 


517 


464 


385 


281 


23 


561 


533 


480 


401 


295 


21 


.82577 


.83549 


.84495 


.85416 


.86310 


20 


593 


565 


511 


431 


325 


19 


610 


581 


526 


446 


340 


18 


626 


597 


542 


461 


354 


17 


643 


613 


557 


476 


369 


16 


.82659 


.83629 


.84573 


.85491 


.86384 


15 


675 


645 


588 


506 


398 


14 


692 


660 


604 


521 


413 


13 


708 


676 


619 


536 


427 


13 


724 


692 


635 


551 


442 


11 


> .82741 


.83708 


.84650 


.85567 


.86457 


10 


757 


724 


666 


582 


471 


9 


773 


740 


681 


597 


486 


8 


790 


756 


697 


612 


501 


7 


806 


772 


712 


627 


515 


6 


,82822 


.83788 


.84728 


.85642 


.86530 


5 


839 


804 


743 


657 


544 


4 


855 


819 


759 


672 


559 


3 


871 


835 


774 


687 


573 


3 


887 


851 


789 


702 


588 


1 


.82904 


.83867 


.84805 


.85717 


.86603 





\ 34° 


33° 


32° 


31° 


30° 


Cos 


145° 


146° 


147° 


148° 


149° 








23 


3 







IV. NATURAL SINES 





119° 


118° 


117° 


116° 


115° 


Sin 


60° 


61° 


63° 


63° 


64° 


0' 


.86603 


.87462 


.88295 


.89101 


.89879 


1 


617 


476 


308 


114 


892 


3 


632 


490 


322 


127 


905 


3 


646 


504 


336 


140 


918 


4 


661 


518 


349 


153 


930 


5 


.86675 


.87532 


.88363 


.89167 


.89943 


6 


690 


546 


377 


180 


956 


7 


704 


561 


390 


193 


968 


8 


719 


575 


404 


206 


981 


9 


733 


589 


417 


219 


994 


10 


.86748 


.87603 


.88431 


.89232 


.90007 


11 


762 


617 


445 


245 


019 


13 


777 


631 


458 


259 


032 


13 


791 


645 


472 


272 


045 


14 


805 


659 


485 


285 


057 


15 


.86820 


.87673 


.88499 


.89298 


.90070 


16 


834 


687 


512 


311 


082 


17 


849 


701 


526 


324 


095 


18 


863 


715 


539 


337 


108 


19 


878 


729 


553 


350 


120 


20 


.86892 


.87743 


.88566 


.89363 


.90133 


31 


906 


756 


580 


376 


146 


33 


921 


770 


593 


389 


158 


33 


935 


784 


607 


402 


171 


34 


949 


798 


620 


415 


183 


35 


.86964 


.87812 


.88634 


.89428 


.90196 


36 


978 


826 


647 


441 


208 


37 


993 


840 


661 


454 


221 


38 


.87007 


854 


674 


467 


233 ! 


39 


021 


868 


688 


480 


246 


30 


.87036 


.87882 


.88701 


.89493 


.90259 


31 


050 


896 


715 


506 


271 , 


33 


064 


909 


728 


519 


284 


33 


079 


923 


741 


532 


296 


34 


093 


937 


755 


545 


309 


35 


.87107 


.87951 


.88768 


.89558 


.90321 


36 


121 


965 


782 


571 


334 


37 


136 


979 


795 


584 


346 i 


38 


150 


993 


808 


597 


358 1 


39 


164 


.88006 


• 822 


610 


371 


40 


.87178 


.88020 


.88835 


.89623 


.90383 i 


41 


193 


034 


848 


636 


396 


43 


207 


048 


862 


649 


408 1 


43 


221 


062 


875 


662 


421 


44 


235 


075 


888 


674 


433 i 


45 


.87250 


.88089 


.88902 


.89687 


.90446 


46 


264 


103 


915 


700 


458 i 


47 


278 


117 


928 


713 


470 


48 


292 


130 


942 


726 


483 


49 


306 


144 


955 


739 


495 


50 


.87321 


.88158 


.88968 


.89752 


.90507 


51 


335 


172 


981 


764 


520 


53 


349 


185 


995 


777 


532 ; 


53 


363 


199 


.89008 


790 


545 


54 


377 


213 


021 


803 


557 i 


55 


.87391 


.88226 


.89035 


.89816 


.90569 


56 


406 


240 


048 


828 


582 1 


57 


420 


254 


061 


841 


594 


58 


434 


267 


074 


854 


606 1 


59 


448 


281 


087 


867 


618 1 


60 


.87462 


.88295 


.89101 


.89879 


.90631 i 




39° 


38° 


37° 


36° 


35° 


Cos 


150° 


151° 


152° 


153° 


154° 






2 


34 







AND COSINES 



114° 


113° 


112° 1 


111° 


110° 


Sin 


65° 


66° 


67° 


68° 


69° 




.90631 


.91355 


.92050 


.92718 


.93358 


60' 


643 


366 


062 


729 


368 


59 


655 


378 


073 


740 


379 


58 


668 


390 


085 


751 


389 


57 


680 


402 


096 


762 


400 


56 


.90692 


.91414 


.92107 


.92773 


.93410 


55 


704 


425 


119 


784 


420 


54 


717 


437 


130 


794 


431 


53 


729 


449 


141 


805 


441 


53 


741 


461 


152 


816 


452 


51 


.90753 


.91472 


.92164 


.92827 


.93462 


50 


766 


484 


175 


838 


472 


49 


778 


496 


186 


849 


483 


48 


790 


508 


198 


859 


493 


47 


802 


519 


209 


870 


503 


46 


.90814 


.91531 


.92220 


.92881 


.93514 


45 


826 


543 


231 


892 


524 


44 


839 


555 


243 


902 


534 


43 


851 


566 


254 


913 


544 


43 


863 


578 


265 


924 


555 


41 


.90875 


.91590 


.92276 


.92935 


.93565 


40 


887 


601 


287 


945 


575 


39 


899 


613 


299 


956 


585 


38 


911 


625 


310 


967 


596 


37 


924 


636 


321 


978 


606 


36 


.90936 


.91648 


.92332 


.92988 


.93616 


35 


948 


660 


343 


999 


626 


34 


960 


671 


355 


.93010 


637 


33 


972 


683 


366 


020 


647 


33 


984 


694 


377 


031 


657 


31 


.90996 


.91706 


.92388 


.93042 


.93667 


30 


.91008 


718 


399 


052 


677 


39 


020 


729 


410 


063 


688 


38 


032 


741 


421 


•074 


698 


37 


044 


752 


432 


084 


708 


36 


.91056 


.91764 


.92444 


.93095 


.93718 


35 


068 


775 


455 


106 


728 


34 


080 


787 


466 


116 


738 


33 


092 


799 


477 


127 


748 


33 


104 


810 


488 


137 




31 


.91116 


.91822 


.92499 


.93148 


.93769 


30 


128 


833 


510 


159 


779 


19 


140 


845 


521 


169 


789 


18 


152 


856 


532 


180 


799 


17 


164 


868 


543 


190 


809 


16 


.91176 


.91879 


.92554 


.93201 


.93819 


15 


188 


891 


565 


211 


829 


14 


200 


902 


576 


222 


839 


13 


212 


914 


587 


232 


849 


13 


224 


925 


598 


243 


859 


11 


.91236 


.91936 


.92609 


.93253 


.93869 


10 


^ 248 


948 


620 


264 


879 


9 


260 


959 


631 


274 


889 


8 


272 


971 


642 


285 


899 


7 


283 


982 


653 


295 


909 


6 


.91295 


.91994 


.92664 


.93306 


.93919 


5 


307 


.92005 


675 


316 


929 


4 


319 


016 


686 


327 


939 


3 


331 


028 


697 


337 


949 


3 


343 


039 


707 


348 


959 


1 


.91355 


.92050 


.92718 


.93358 


.93969 





J 34° 


33° 


33° 


31° 


30° 


Cos 


155° 


156° 


157° 


158° 


159° 





235 



IV. NATURAL SINES 





109° 


108° 


107° 


106° 


105° 


Sin 


70° 


71° 


73° 


73° 


74° 


0' 


.93969 


.94552 


.95106 


.95630 


.96126 


1 


979 


561 


115 


639 


134 


2 


989 


571 


124 


647 


142 


3 


999 


580 


133 


656 


150 


4 


.94009 


590 


142 


664 


158 


5 


.94019 


.94599 


.95150 


.95673 


.96166 


6 


029 


609 


159 


681 


174 


7 


039 


618 


168 


690 


182 


8 


049 


627 


177 


698 


190 


9 


058 


637 


186 


707 


198 


10 


.94068 


.94646 


.95195 


.95715 


.96206 


11 


078 


656 


204 


724 


214 


13 


088 


665 


213 


732 


222 


13 


098 


674 


222 


740 


230 


14 


108 


684 


231 


749 


238 


15 


.94118 


.94693 


.95240 


.95757 


.96246 


16 


127 


702 


248 


766 


253 


17 


137 


712 


257 


774 


261 


18 


147 


721 


266 


782 


269 


19 


157 


730 


275 


791 


277 


20 


.94167 


.94740 


.95284 


.95799 


.96285 


31 


176 


749 


293 


807 


293 


33 


186 


758 


301 


816 


301 


33 


196 


768 


310 


824 


308 


34 


206 


777 


319 


832 


316 


35 


.94215 


.94786 


.95328 


.95841 


.96324 


36 


225 


795 


337 


849 


332 


37 


235 


805 


345 


857 


340 


38 


245 


814 


354 


865 


347 


39 


254 


823 


363 


874 


355 


30 


.94264 


.94832 


.95372 


.95882 


.96363 


31 


274 


842 


380 


890 


371 


33 


284 


851 


389 


898 


379 


33 


293 


860 


398 


907 


386 


34 


303 


869 


407 


915 


394 


35 


.94313 


.94878 


.95415 


.95923 


.96402 


36 


322 


888 


424 


931 


410 


37 


332 


897 


433 


940 


417 


38 


342 


906 


441 


948 


425 


39 


351 


9^15 


450 


956 


433 


40 


.94361 


.94924 


.95459 


.95964 


.96440 


41 


370 


933 


467 


972 


448 


43 


380 


943 


476 


981 


456 


43 


390 


952 


485 


989 


463 


44 


399 


961 


493 


997 


471 


45 


.94409 


.94970 


.95502 


.96005 


.96479 


46 


418 


979 


511 


013 


486 


47 


428 


988 


519 


021 


494 


48 


438 


997 


528 


029 


502 


49 


447 


.95006 


636 


. 037 


509 


50 


.94457 


.95015 


.95545 


.96046 


.96517 


51 


466 


024 


554 


054 


524 


53 


476 


033 


562 


062 


532 


53 


485 


043 


571 


070 


540 


64 


495 


052 


579 


078 


547 


55 


.94504 


.95061 


.95588 


.96086 


.96555 


56 


514 


070 


596 


094 


562 


57 


523 


079 


605 


102 


570 


58 


533 


088 


613 


110 


578 


59 


542 


097 


622 


118 


585 


60 


.94552 


.95106 


.95630 


.96126 


.96593 




19° 


18° 


17° 


16° 


~T¥~~ 


Cos 


160° 


161° 


162° 


163° 


164° 



236 



AND COSINES 



104° 


103° 


102° 


101° 


100° 


Sin 


75° 


76° 


77° 


78° 


79° 




.96593 


.97030 


.97437 


.97815 


.98163 


60' 


600 


037 


444 


821 


168 


59 


608 


044 


450 


827 


174 


58 


615 


051 


457 


833 


179 


57 


623 


058 


463 


839 


185 


56 


.96630 


.97065 


.97470 


.97845 


.98190 


55 


638 


072 


476 


851 


196 


54 


645 


079 


483 


857 


201 


53 


653 


086 


489 


863 


207 


52 


660 


093 


496 


869 


212 


51 


.96667 


.97100 


.97502 


.97875 


.98218 


50 


675 


106 


508 


881 


223 


49 


682 


113 


515 


887 


229 


48 


690 


120 


521 


893 


234 


47 


697 


127 


528 


899 


240 


46 


.96705 


.97134 


.97534 


.97905 


.98245 


45 


712 


141 


541 


910 


250 


44 


719 


148 


547 


916 


256 


43 


727 


155 


553 


922 


261 


43 


734 


162 


560 


928 


267 


41 


.96742 


.97169 


.97566 


.97934 


.98272 


40 


749 


176 


573 


940 


277 


39 


756 


182 


579 


946 


283 


38 


764 


189 


685 


952 


288 


37 


771 


196 


592 


958 


294 


36 


.96778 


.97203 


.97598 


.97963 


.98299 


35 


786 


210 


604 


969 


304 


34 


793 


217 


611 


975 


310 


33 


800 


223 


617 


981 


315 


33 


807 


230 


623 


987 


320 


31 


.96815 


.97237 


.97630 


.97992 


.98325 


30 


822 


244 


636 


998 


331 


29 


829 


251 


642 


.98004 


336 


38 


837 


257 


648 


pio 


341 


37 


844 


264 


655 


016 


347 


36 


.96851 


.97271 


.97661 


.98021 


.98352 


35 


858 


278 


667 


027 


357 


34 


866 


284 


673 


033 


362 


33 


873 


291 


680 


039 


368 


33 


880 


298 


686 


044 


373 


31 


.96887 


.97304 


.97692 


.98050 


.98378 


30 


894 


311 


698 


056 




19 


902 


318 


705 


061 


389 


18 


909 


325 


711 


067 


394 


17 


916 


331 


717 


073 


399 


16 


.96923 


.97338 


.97723 


.98079 


.98404 


15 


930 


345 


729 


084 


409 


14 


937 


351 


735 


090 


414 


13 


945 


358 


742 


096 


420 


13 


952 


365 


748 


101 


425 


11 


.96959 


.97371 


.97754 


.98107 


.98430 


10 


966 


378 


760 


112 


435 


9 


973 


384 


766 


118 


440 


8 


980 


391 


772 


124 


445 


7 


987 


398 


778 


129 


450 


6 


.96994 


.97404 


.97784 


.98135 


.98455 


5 


.97001 


411 


791 


140 


461 


4 


008 


417 


797 


146 


466 


3 


015 


424 


803 


152 


471 


3 


023 


430 


809 


157 


476 


1 


.97030 


.97437 


.97815 


.98103 


.98481 





14° 


13° 


13° 


11° 


10° 


Cos 


165° 


166° 


167° 


168° 


169° 





237 



IV. NATURAL SINES 





99° 


98° 


97° 


96° 


95° 


Sin 


80° 


81° 


83° 


83° 


84° 


(y 


.98481 


.98769 


.99027 


.99255 


.99452 


1 


486 


773 


031 


258 


455 


2 


491 


778 


035 


262 


458 


3 


496 


782 


039 


265 


461 


4 


501 


787 


043 


269 


464 


5 


.98506 


.98791 


.99047 


.99272 


.99467 


6 


511 


796 


051 


276 


470 


7 


516 


800 


055 


279 


473 


8 


521 


805 


059 


283 


476 


9 


526 


809 


063 


286 


479 


10 


.98531 


.98814 


.99067 


.99290 


.99482 


11 


536 


818 


071 


293 


485 . 


13 


541 


823 


075 


297 


488 


13 


546 


827 


079 


300 


491 


14 


551 


832 


083 


303 


494 


15 


.98556 


.98836 


.99087 


.99307 


.99497 


16 


561 


841 


091 


310 


500 


17 


565 


845 


094 


314 


503 


18 


570 


849 


098 


317 


506 


19 


575 


854 


102 


320 


508 


20 


.98580 


.98858 


.99106 


.99324 


.99511 


31 


585 


863 


110 


327 


514 


33 


590 


867 


114 


331 


517 


33 


595 


871 


118 


334 


520 


34 


600 


876 


122 


337 


523 


35 


.98604 


.98880 


.99125 


.99341 


.99526 


36 


609 


884 


129 


344 


528 


37 


614 


889 


133 


347 


531 


38 


619 


893 


137 


351 


534 


39 


624 


897 


141 


354 


537 


30 


.98629 


.98902 


.99144 


.99357 


.99540 


31 


633 


906 


148 


360 


542 


33 


638 


910 


152 


364 


545 


33 


643 


914 


156 


367 


548 


34 


648 


919 


160 


370 


551 


35 


.98652 


.98923 


.99163 


.99374 


.99553 


36 


657 


927 


167 


377 


556 


37 


662 


931 


171 


380 


559 


38 


667 


936 


175 


383 


562 


39 


671 


940 


178 


386 


564 


40 


.98676 


.98944 


.99182 


.99390 


.99567 


41 


681 


948 


186 


393 


570 


43 


686 


953 


189 


396 


572 


43 


690 


957 


193 


399 


575 


44 


695 


961 


197 


402 


578 


45 


.98700 


.98965 


.99200 


.99406 


.99580 


46 


704 


969 


204 


409 


583 


47 


709 


973 


208 


412 


586 


48 


714 


978 


211 


415 


588 


49 


718 


982 


215 


418 


591 


50 


.98723 


.98986 


.99219 


.99421 


.99594 


51 


728 


990 


222 


424 


596 


53 


732 


994 


226 


428 


599 


53 


737 


998 


230 


431 


602 


64 


. 741 


.99002 


233 


434 


604 


55 


.98746 


.99006 


.99237 


.99437 


.99607 


56 


751 


on 


240 


440 


609 


57 


755 


015 


244 


443 


612 




760 


019 


248 


446 


614 


59 


764 


023 


251 


449 


617 


60 


.98769 


.99027 


.99255 


.99452 


.99619 




9° 


8° 
171° 


7° 


6° 


5° 


Cos 


170° 


172° 


173° 


174° 



238 



AND COSINES 



94° 


93° 


92° 


91° 


90° 


Sin 


85° 


86° 


87° 


88° 


89° 




.99619 


.99756 


.99863 


.99939 


«99985 


60' 


622 


758 


864 


940 


985 


59 


625 


760 


866 


941 


986 


58 


627 


762 


867 


942 


986 


57 


630 


764 


869 


943 


987 


56 


.99632 


.99766 


.99870 


.99944 


.99987 


55 


635 


768 


872 


945 


988 


54 


637 


770 


873 


946 


988 


53 


639 


772 


875 


947 


989 


53 


642 


774 


876 


948 


989 


51 


.99644 


.99776 


.99878 


.99949 


.99989 


50 


647 


778 


879 


950 


990 


49 


649 


780 


881 


951 


990 


48 


652 


782 


882 


952 


991 


47 


654 


784 


883 


952 


991 


46 


.99657 


.99786 


.99885 


.99953 


.99991 


45 


659 


788 


886 


954 


992 


44 


661 


790 


888 


955 


992 


43 


664 


792 


889 


956 


993 


43 


666 


793 


890 


957 


993 


41 


.99668 


.99795 


.99892 


.99958 


.99993 


40 


671 


797 


893 


959 


994 


39 


673 


799 


894 


959 


994 


38 


676 


801 


896 


960 


994 


37 


678 


803 


897 


961 


995 


36 


.99680 


.99804 


.99898 


.99962 


.99995 


35 


683 


806 


900 


963 


995 


34 


685 


808 


901 


963 


995 


33 


687 


810 


902 


964 


996 


33 


689 


812 


904 


965 


996 


31 


.99692 


.99813 


.99905 


.99966 


.99996 


30 


694 


815 


906 


966 


996 


39 


696 


817 


907 


967 


997 


38 


699 


819 


909 


968 


997 


37 


701 


821 


910 


969 


997 


36 


.99703 


.99822 


.99911 


.99969 


.99997 


35 


705 


824 


912 


970 


998 


34 


708 


826 


913 


971 


998 


33 


710 


827 


915 


972 


998 


33 


712 


829 


916 


972 


998 


31 


.99714 


.99831 


.99917 


.99973 


.99998 


30 


716 


833 


918 


974 


998 


19 


719 


834 


919 


974 


999 


18 


721 


836 


921 


975 


999 


17 


723 


838 


922 


976 


999 


16 


.99725 


.99839 


.99923 


.99976 


.99999 


15 


727 


841 


924 


977 


999 


14 


729 


842 


925 


977 


999 


13 


731 


844 


926 


978 


999 


13 


734 


846 


927 


979 


999 


11 


.99736 


.99847 


.99929 


.99979 


1.00000 


10 


738 


849 


930 


980 


000 


9 


740 


851 


931 


980 


000 


8 


742 


852 


932 


981 


000 


7 


744 


854 


933 


982 


000 


6 


.99746 


.99855 


.99934 


.99982 


1 .00000 


5 


748 


857 


935 


983 


000 


4 


750 


858 


936 


983 


000 


3 


752 


860 


937 


984 


000 


2 


754 


861 


938 


984 


000 


1 


.99756 


-99863 


.99939 


.99985 


1.00000 





4° 


3° 


2° 


1° 


0° 


Cos 


175° 


176° 


177° 


178° 


179° 





23t, 



V. NATURAL TANGENTS 





179° 


178° 


177° 


176° 


175° 


Tan 


0° 


1° 


2° 


3° 


4° 


(y 


.00000 


.01746 


.03492 


.05241 


.06993 


1 


029 


775 


521 


270 


.07022 


2 


058 


804 


550 


299 


051 


3 


087 


833 


579 


328 


080 


4 


lib 


862 


609 


357 


110 


5 


.00145 


.01891 


.03638 


.05387 


.07139 


6 


175 


920 


667 


416 


168 


7 


204 


949 


696 


445 


197 


8 


233 


978 


725 


474 


227 


9 


262 


.02007 


754 


503 


256 


10 


.00291 


.02036 


.03783 


.05533 


.07285 


11 


320 


066 


812 


562 


314 


13 


349 


095 


842 


591 


344 


13 


378 


124 


871 


620 


373 


14 


407 


153 


900 


649 


402 


15 


.00436 


.02182 


.03929 


.05678 


.07431 


16 


465 


211 


958 


708 


461 


17 


495 


240 


987 


737 


490 


18 


524 


269 


.04016 


766 


519 


19 


553 


298 


046 


795 


548 


20 


.00582 


.02328 


.04075 


.05824 


.07578 


21 


611 


357 


104 


854 


607 


22 


640 


386 


133 


883 


636 


23 


669 


415 


162 


912 


665 


24 


698 


444 


191 


941 


695 


25 


.00727 


.02473 


.04220 


.05970 


.07724 


26 


756 


502 


250 


999 


753 


27 


785 


531 


279 


.06029 


782 


28 


815 


560 


308 


058 


812 


29 


844 


589 


337 


087 


841 


30 


.00873 


.02619 


.04366 


.06116 


.07870 


31 


902 


648 


395 


145 


899 


32 


931 


677 


424 


175 


929 


33 


960 


706 


454 


204 


958 


34 


989 


735 


483 


233 


987 


35 


.01018 


.02764 


.04512 


.06262 


.08017 


36 


047 


793 


541 


291 


046 


37 


076 


822 


570 


321 


075 


38 


105 


851 


599 


350 


104 


39 


135 


881 


628 


379 


134 


40 


.01164 


.02910 


.04658 


.06408 


.08163 


41 


193 


939 


687 


437 


192 


42 


222 


968 


716 


467 


221 


43 


251 


997 


745 


496 


251 


44 


280 


.03026 


774 


525 


280 


45 


.01309 


.03055 


.04803 


.06554 


.08309 


46 


338 


084 


833 


584 


339 


47 


367 


114 


862 


613 


368 


48 


396 


143 


891 


642 


397 


49 


425 


172 


920 


671 


427 


50 


.01455 


.03201 


.04949 


.06700 


.08456 


51 


484 


230 


978 


730 


485 


52 


513 


259 


.05007 


759 


514 


53 


542 


288 


037 


788 


544 


54 


571 


317 


066 


817 


573 


55 


.01600 


.03346 


.05095 


.06847 


.08602 


56 


629 


376 


124 


876 


632 


57 


658 


405 


153 


905 


661 


58 


687 


434 


182 


934 


690 


59 


716 


463 


212 


963 


720 


60 


.01746 


.03492 


.05241 


.06993 


.08749 




89° 


88° 


87° 


86° 


85° 


Cot 


90° 


91° 


92° 


93° 


94° 



240 



AND COTANGENTS 



174° 


173° 


172° 


171° 


170° 


Tan 


5° 


6° 


7° 


8° 


9° 




.08749 


.10510 


.12278. 


.14054 


.15838 


60' 


778 


540 


308 


084 


868 


59 


807 


569 


338 


113 


898 


58 


837 


599 


367 


143 


928 


57 


866 


628 


397 


173 


958 


56 


.08895 


.10657 


.12426 


.14202 


.15988 


55 


925 


687 


456 


232 


.16017 


54 


954 


716 


485 


262 


047 


53 


983 


746 


515 


291 


077 


53 


.09013 


775 


544 


321 


107 


51 


.09042 


.10805 


.12574 


.14351 


.16137 


50 


071 


834 


603 


381 


167 


49 


101 


863 


633 


410 


196 


48 


130 


893 


662 


440 


226 


47 


159 


922 


692 


470 


256 


46 


.09189 


.10952 


.12722 


.14499 


.16286 


45 


218 


981 


751 


529 


316 


44 


247 


.11011 


781 


559 


346 


43 


277 


040 


810 


588 


376 


43 


306 


070 


840 


618 


405 


41 


.09335 


.11099 


.12869 


.14648 


.16435 


40 


365 


128 


899 


678 


465 


39 


394 


158 


929 


707 


495 


38 


423 


187 


958 


737 


525 


37 


453 


217 


988 


767 


555 


36 


.09482 


.11246 


.13017 


.14796 


.16585 


35 


511 


276 


047 


826 


615 


34 


541 


305 


076 


856 


645 


33 


570 


335 


106 


886 


674 


33 


600 


364 


136 


915 


704 


31 


.09629 


.11394 


.13165 


.14945 


.16734 


30 


658 


423 


195 


975 


764 




688 


452 


224 


.15005 


794 


38 


717 


482 


254 


034 


824 


37 


746 


511 


284 


064 


854 


36 


.09776 


.11541 


.13313 


.15094 


.16884 


35 


805 


570 


343 


124 


914 


34 


834 


600 


372 


153 


944 


33 


864 


629 


402 


183 


974 


33 


893 


659 


432 


213 


.17004 


31 


.09923 


.11688 


.13461 


.15243 


.17033 


30 


952 


718 


491 


272 


063 


19 


981 


747 


521 


302 


093 


18 


.10011 


777 


550 


332 


123 


17 


040 


806 


580 


362 


153 


16 


.10069 


.11836 


.13609 


.15391 


.17183 


15 


099 


865 


639 


421 


213 


14 


128 


895 


669 


451 


243 


13 


158 


924 


698 


481 


273 


13 


187 


954 


728 


511 


303 


11 


.10216 


.11983 


.13758 


.15540 


.17333 


10 


246 


.12013 


787 


570 


363 


9 


275 


042 


817 


600 


393 


8 


305 


072 


846 


630 


423 


7 


334 


101 


876 


660 


453 


6 


.10363 


.12131 


.13906 


.15689 


.17483 


5 


393 


160 


935 


719 


513 


4 


422 


190 


965 


749 


543 


3 


452 


219 


995 


779 


573 


3 


481 


249 


.14024 


809 


603 


1 


.10510 


.12278 


. 14054 


.15838 


.17633 





84° 


83° 


8*^° 


81° 


80° 


Cot 


95° 


96° 


97° 


98° 


99° 





241 



V. NATURAL TANGENTS 





169° 


168° 


167° 


166° 


165° 


Tan 


10° 


11° 


13° 


13° 


14° 


0' 


.17633 


.19438 


.21256 


.23087 


.24933 


1 


663 


468 


286 


117 


964 


3 


693 


498 


316 


148 


995 


3 


723 


529 


347 


179 


.25026 


4 


753 


559 


377 


209 


056 


5 


.17783 


.19589 


.21408 


.23240 


.25087 


6 


813 


619 


438 


271 


118 


7 


843 


649 


469 


301 


149 


8 


873 


680 


499 


332 


180 


9 


903 


710 


529 


363 


211 


10 


.17933 


.19740 


.21560 


.23393 


.25242 


11 


963 


770 


590 


424 


273 


13 


993 


801 


621 


455 


304 


13 


.18023 


831 


651 


485 


335 


14 


053 


861 


682 


516 


366 


15 


.18083 


.19891 


.21712 


.23547 


.25397 


16 


113 


921 


743 


578 


428 


17 


143 


952 


773 


608 


459 


18 


173 


982 


804 


639 


490 


19 


203 


.20012 


834 


670 


521 


30 


.18233 


.20042 


.21864 


.23700 


.25552 


31 


263 


073 


895 


731 


583 


33 


293 


103 


925 


762 


614 


33 


323 


133 


956 


793 


645 


34 


353 


164 


986 


823 


676 


35 


.18384 


.20194 


.22017 


.23854 


.25707 


36 


414 


224 


047 


885 


738 


37 


444 


254 


078 


916 


769 


38 


474 


285 


108 


946 


800 


39 


504 


315. 


139 


977 


831 


30 


.18534 


.20345 


.22169 


.24008 


.25862 


31 


564 


376 


200 


039 


893 


33 


594 


406 


231 


069 


924 


33 


624 


436 


261 


100 


955 


34 


654 


466 


292 


131 


986 


35 


.18684 


.20497 


.22322 


.24162 


.26017 


36 


714 


527 


353 


193 


048 


37 


745 


557 


383 


223 


079 


38 


775 


588 


414 


254 


110 


39 


805 


618 


444 


285 


141 


40 


.18835 


.20648 


.22475 


.24316 


.26172 


41 


865 


679 


505 


347 


203 


43 


895 


709 


536 


377 


235 


43 


925 


739 


567 


408 


266 


44 


955 


770 


597 


439 


297 


45 


.18986 


.20800 


.22628 


.24470 


; 26328 


46 


.19016 


830 


658 


oOl 


359 


47 


046 


861 


689 


532 


390 


48 


076 


891 


719 


562 


421 


49 


106 


921 


750 


593 


452 


60 


.19136 


.20952 


.22781 


.24624 


.26483 


61 


166 


982 


811 


655 


515 


63 


197 


.21013 


842 


686 


546 


63 


227 


043 


872 


717 


577 


64 


257 


073 


903 


747 


608 


55 


.19287 


.21104 


.22934 


.24778 


.26639 


66 


317 


134 


964 


809 


670 


67 


347 


164 


995 


840 


701 


68 


378 


195 


.23026 


871 


733 


69 


408 


225 


056 


902 


764 


60 


.19438 


.21256 


.23087 


.24933 


.26795 




79° 


78° 


77° 


76° 


75° 


Cot 


100° 


101° 


102° 


103° 


104° 



242 



AND COTANGENTS 



164° 


163° 


162° 


161° 


160° 


Tan 


15° 


16° 


17° 


18° 


19° 




.26795 


.28675 


.30573 


.32492 


.34433 


60' 


826 


706 


605 


524 


465 


59 


857 


738 


637 


556 


498 


58 


888 


769 


669 


588 


530 


57 


920 


801 


700 


621 


563 


56 


.26951 


.28832 


.30732 


.32653 


.34596 


55 


982 


864 


764 


685 


628 


54 


.27013 


895 


796 


717 


661 


53 


044 


927 


828 


749 


693 


52 


076 


958 


860 


782 


726 


51 


.27107 


.28990 


.30891 


.32814 


.34758 


50 


138 


.29021 


923 


846 


791 


49 


169 


053 


955 


878 


824 


48 


201 


084 


987 


911 


856 


47 


232 


116 


.31019 


943 


889 


46 


.27263 


.29147 


.31051 


.32975 


.34922 


45 


294 


179 


083 


.33007 


954 


44 


326 


210 


115 


040 


987 


43 


357 


242 


147 


072 


.35020 


43 


388 


274 


178 


104 


052 


41 


.27419 


.29305 


.31210 


.33136 


.35085 


40 


451 


337 


242 


169 


118 


39 


482 


368 


274 


201 


150 


38 


513 


400 


306 


233 


183 


37 


545 


432 


338 


266 


216 


36 


.27576 


.29463 


.31370 


.33298 


.35248 


35 


607 


495 


402 


330 


281 


34 


638 


526 


434 


363 


314 


33 


670 


558 


466 


395 


346 


33 


701 


590 


498 


427 


379 


31 


.27732 


.29621 


.31530 


.33460 


.35412 


30 


764 


653 


562 


492 


445 


39 


795 


685 


594 


524 


477 


38 


826 


716 


626 


557 


510 


37 


858 


748 


658 


589 


543 


36 


.27889 


.29780 


.31690 


.33621 


.35576 


35 


921 


811 


722 


654 


608 


34 


952 


843 


754 


686 


641 


33 


983 


875 


786 


718 


674 


33 


.28015 


906 


818 


751 


707 


31 


.28046 


.29938 


.31850 


.33783 


.35740 


30 


077 


970 


882 


816 


772 


19 


109 


.30001 


914 


848 


805 


18 


140 


033 


946 


881 


838 


17 


172 


065 


978 


913 


871 


16 


.28203 


.30097 


.32010 


.33945 


.35904 


15 


234 


128 


042 


978 


937 


14 


266 


160 


074 


.34010 


969 


13 


297 


192 


106 


043 


.36002 


13 


329 


224 


139 


075 


035 


11 


.28360 


.30255 


.32171 


.34108 


.36068 


10 


391 


287 


203 


140 


101 


9 


423 


319 


235 


173 


134 


8 


454 


351 


267 


205 


167 


7 


486 


382 


299 


238 


199 


6 


.28517 


.30414 


.32331 


.34270 


.36232 


5 


549 


446 


363 


303 


265 


4 


580 


478 


396 


335 


298 


3 


612 


509 


428 


368 


331 


3 


643 


541 


460 


400 


364 


1 


.28675 


.30573 


.32492 


.34433 


.36397 





74° 


73° 


73^ 


71° 


70° 


Cot 


105° 


106° 


107° 


108° 


109° 





243 



V. NATURAL TANGENTS 



L. 





159° 


158° 


1 157° 


1 156° 


155° 


Tan 


30° 


31" 


3-4° 


33° 


34° 


<y 


.36397 


.38386 


.40403 


.42447 


.44523 


1 


430 


420 


436 


482 


558 


2 


463 


453 


470 


516 


593 


3 


496 


487 


504 


551 


627 


4 


529 


520 


538 


585 


662 


5 


.36562 


.38553 


.40572 


.42619 


.44697 


6 


595 


587 


606 


654 


732 


7 


628 


620 


640 


688 


767 


8 


661 


654 


674 


722 


802 


9 


694 


687 


707 


757 


837 


10 


.36727 


.38721 


.40741 


.42791 


.44872 


11 


760 


754 


775 


826 


907 


13 


793 


787 


809 


860 


942 


13 


826 


821 


843 


894 


977 


14 


859 


854 


877 


929 


.45012 


15 


.36892 


.38888 


.40911 


.42963 


.45047 


16 


925 


921 


945 


998 


082 


}7 


958 


955 


979 


.43032 


117 


18 


991 


988 


.41013 


067 


152 


19 


.37024 


.39022 


047 


101 


187 


20 


.37057 


.39055 


.41081 


.43136 


.45222 


31 


090 


089 


115 


170 


257 


33 


123 


122 


149 


205 


292 


33 


157 


156 


183 


239 


327 


34 


190 


190 


217 


274 


362 


35 


.37223 


.39223 


.41251 


.43308 


.45397 


36 


256 


257 


285 


343 


432 


37 


289 


290 


319 


378 


467 


38 


322 


324 


353 


412 


502 


39 


355 


357 


387 


447 


538 


30 


.37388 


.39391 


.41421 


.43481 


.45573 , 


31 


422 


425 


455 


516 


608 ; 


33 


455 


458 


490 


550 


643 ! 


33 


488 


492 


524 


585 


678 ■ 


34 


521 


526 


558 


620 


713 


35 


.37554 


.39559 


.41592 


.43654 


.45748 i 


36 


588 


593 


626 


689 


784 : 


37 


621 


626 


660 


724 


819 


38 


654 


660 


694 


758 


854 


39 


687 


694 


728 


793 


8<^9 


40 


.37720 


.39727 


.41763 


.43828 


.45924 


41 


754 


761 


797 


862 


960 


43 


787 


795 


831 


897 


995 


43 


820 


829 


865 


932 


.46030 


44 


853 


862 


899 


966 


065 


45 


.37887 


.39896 


.41933 


.44001 


.46101 


46 


920 


930 


968 


036 


136 


47 


953 


963 


.42002 


071 


171 


48 


986 


997 


036 


105 


206 


49 


.38020 


.40031 


070 


140 


242 


60 


.38053 


.40065 


.42105 


.44175 


.46277 


51 


086 


098 


139 


210 


312 


53 


120 


132 


173 


244 


348 


53 


153 


166 


207 


279 


383 


54 


180 


200 


242 


314 


418 


55 


.38220 


.40234 


.42276 


.44349 


.46454 j 


56 


253 


267 


310 


384 


489 1 


57 


286 


301 


345 


418 


525 


58 


320 


335 


379 


453 


560 


59 


353 


369 


413 


488 


595 , 


60 


.38386 


.40403 


.42447 


.44523 


.46631 ' 




69° 


68° 


67° 


66° 


65° 


Cot 


110° 


111° 


112° 


113° 


114° 



244 



AND COTANGENTS 



154° 


153° 


152° 


151° 


150° 


Tan 


25° 


36° 


27° 


38° 


39° 




.46631 


.48773 


.50953 


.53171 


.55431 


6^ 


666 


809 


989 


208 


469 


59 


702 


845 


.51026 


246 


507 


58 


737 


881 


063 


283 


545 


57 


772 


917 


099 


320 


583 


56 


.46808 


.48953 


.51136 


.53358 


.55621 


55 


843 


989 


173 


395 


659 


54 


879 


.49026 


209 


432 


697 


53 


914 


062 


246 


470 


736 


53 


950 


098 


283 


507 


774 


51 


.46985 


.49134 


.51319 


.53545 


.55812 


50 


.47021 


170 


356 


5S2 


850 


49 


056 


206 


393 


620 


888 


48 


092 


242 


430 


657 


926 


47 


128 


278 


467 


694 


964 


46 


.47163 


.49315 


.51503 


.53732 


.56003 


45 


199 


351 


540 


769 


041 


44 


234 


387 


577 


807 


079 


43 


270 


423 


614 


844 


117 


43 


305 


459 


651 


882 


156 


41 


.47341 


.49495 


.51688 


.53920 


.56194 


40 


377 


532 


724 


957 


232 


39 


412 


568 


761 


995 


270 


38 


448 


604 


798 


.54032 


309 


37 


483 


640 


835 


070 


347 


36 


.47519 


.49677 


.51872 


.54107 


.56385 


35 


555 


713 


909 


145 


424 


34 


590 


749 


946 


183 


462 


33 


626 


786 


983 


220 


501 


33 


662 


822 


,52020 


258 


539 


31 


.47698 


.49858 


.52057 


.54296 


.56577 


30 


733 


894 


094 


333 


616 


39 


769 


931 


131 


371 


654 


38 


805 


967 


168 


409 


693 


37 


840 


.50004 


205 


446 


731 


36 


.47876 


.50040 


.52242 


.54484 


.56769 


35 


912 


076 


279 


522 


808 


34 


948 


113 


316 


560 


846 


33 


984 


149 


353 


597 


885 


33 


.48019 


185 


390 


635 


923 


31 


.48055 


.50222 


.52427 


.54673 


.56962 


30 


091 


258 


464 


711 


.57000 


19 


127 


295 


501 


748 


039 


18 


163 


331 


538 


786 


078 


17 


198 


368 


575 


824 


116 


16 


.48234 


.50404 


.52613 


.54862 


.57155 


15 


270 


441 


650 


900 


193 


14 


306 


477 


687 


938 


232 


13 


342 


514 


724 


975 


271 


13 


378 


550 


761 


.55013 


309 


11 


.48414 


.50587 


.52798 


.55051 


.57348 


10 


4.50 


623 


836 


089 


386 


9 


486 


660 


873 


127 


425 


8 


521 


696 


910 


165 


464 


7 


557 


733 


947 


203 


503 


6 


.48593 


.50769 


.52985 


.55241 


.57541 


5 


629 


806 


.53022 


279 


580 


4 


665 


843 


059 


317 


619 


3 


701 


879 


096 


355 


657 


3 


737 


916 


134 


393 


696 


1 


.48773 


.50953 


.53171 


.55431 


.57735 





64° 


63° 


63° 


61° 


60° 


Cot 


115° 


116° 


117° 


118° 


119° 





245 







V. 


NATURAL TANGENTS 




U9o^ 


148° 


147° 


146° 


145° 1 


Tan 


30° 


31° 


32° 


33° 


34° 


0' 


.57735 


.60086 


.62487 


.64941 


.67451 


1 


774 


126 


527 


982 


493 


2 


813 


165 


568 


.65024 


636 


3 


851 


205 


608 


065 


578 


4 


890 


245 


649 


106 


620 


5 


.57929 


.60284 


.62689 


.65148 


.67663 


6 


968 


324 


730 


189 


705 




.58007 


364 


770 


231 


748 


Q 


046 


403 


811 


272 


790 


9 


085 


443 


852 


314 


832 


10 


.58124 


.60483 


.62892 


.65355 


.67875 


11 


162 


522 


933 


397 


917 


12 


201 


562 


973 


438 


960 


13 


240 


602 


.63014 


480 


.68002 


14 


279 


642 


055 


521 


045 


15 




.60681 


.63095 


.65563 


.68088 


16 


357 


721 


136 


604 


130 


17 


396 


761 


177 


646 


173 


18 


435 


801 


217 


688 


215 


19 


474 


841 


258 


729 


258 


20 


.58513 


.60881 


.63299 


.65771 


.68301 


31 


552 


' 921 


340 


813 


343 


32 


591 


960 


380 


854 


386 


23 


631 


.61000 


421 


896 


429 


24 


670 


040 


462 


938 


471 


25 


.58709 


.61080 


.63503 


.65980 


.68514 


26 


748 


120 


544 


.66021 


557 


27 


787 


160 


584 


063 


600 


28 




200 


625 


105 


642 


29 


865 


240 


666 


147 


685 


30 


.58905 


.61280 


.63707 


.66189 


.68728 


31 


944 


320 


748 


230 


771 


32 


983 


360 


789 


272 


814 


33 


.59022 


400 


830 


314 


857 


34 


061- 


440 


871 


356 


900 


35 


.59101 


.61480 


.63912 


.66398 


.68942 


36 


140 


520 


953 


440 


985 


37 


179 


561 


994 


482 


.69028 


38 


218 


601 


.64035 


524 


071 


39 


258 


641 


076 


566 


114 


40 


.59297 


.61681 


.64117 


.66608 


.69157 


41 


336 


721 


158 


650 


200 


42 


376 


761 


199 


692 


243 


43 


415 


801 


240 


734 


286 


44 


454 


842 


281 


776 


329 


45 


.59494 


.61882 


.64322 


.66818 


.69372 


46 


533 


922 


363 


860 


416 


47 


573 


962 


404 


902 


459 


48 


612 


.62003 


446 


944 


502 


49 


651 


043 


487 


986 


545 


60 


.59691 


.62083 


.64528 


.67028 


.69588 


61 


730 


124 


569 


071 


631 


62 


770 


164 


610 


113 


675 


63 


809 


204 


652 


155 


718 




849 


245 


693 


197 


761 


65 


.59888 


.62285 


.64734 


.67239 


.69804 
847 


66 


928 




775 


282 


67 


967 


366 


817 


324 


891 


68 


.60007 


406 


858 


366 


934 


69 


046 


446 


899 


409 


977 


60 


.60086 


.62487 


.64941 


.67451 


.70021 




59° 


58° 


57° 


56° 


55° 


Cot 


120° 


121<> 


122° 


123° 


124° 



246 



AND COTANGENTS 



144° 


143° 


142° 


141° 


140° 


■ ■ Tan " 


35° 


36° 


37° 


38° 


39° 




.70021 


.72654 


.75355 


.78129 


.80978 


ec 


064 


699 


401 


175 


.81027 


69 


107 


743 


447 


222 


075 


68 


151 


788 


492 


269 


123 


67 


194 


832 


538 


316 


171 


66 


.70238 


.72877 


,75584 


.78363 


.81220 


65 


281 


921 


629 


410 


268 


64 


325 


966 


675 


457 


316 


63 


368 


.73010 


721 


504 


364 


63 


412 


055 


767 


551 


413 


61 


.70455 


.73100 


.75812 


.78598 


.81461 


60 


499 


144 


858 


645 


510 


49 


542 


189 


904 


692 


558 


48 


586 


234 


950 


739 


606 


47 


629 


278 


996 


786 


655 


46 


.70673 


.73323 


.76042 


.78834 


.81703 


46 


717 


368 


088 


881 


752 


44 


760 


413 


134 


928 


800 


43 


804 


457 


180 


975 


849 


43 


848 


502 


226 


.79022 


898 


41 


.70891 


.73547 


.76272 


.79070 


.81946 


40 


935 


592 


318 


117 


995 


39 


979 


637 




164 


.82044 


38 


.71023 


681 


410 


212 


092 


37 


066 


726 


456 


259 


141 


36 


.71110 


.73771 


.76502 


.79306 


.82190 


35 


154 


816 


548 


354 


238 


84 


198 


861 


594 


401 


287 


33 


242 


906 


640 


449 




33 


285 


951 


686 


496 


385 


31 


.71329 


.73996 


.76733 


.79544 


.82434 


30 


373 


.74041 


779 


591 


483 


29 


417 


086 


825 


639 


631 


38 


461 


131 


871 


686 


680 


37 


505 


176 


918 


734 


629 


26 


.71549 


.74221 


.76964 


.79781 


.82678 


25 


593 


267 


.77010 


829 


727 


24 


637 


312 


057 


877 


776 


23 


681 


357 


103 


924 




23 


725 


402 


149 


972 


874 


31 


.71769 


.74447 


.77196 


.80020 


.82923 


30 


813 


492 


242 


067 


972 


19 


857 


538 


289 


115 


.83022 


18 


901 


583 


335 


163 


071 


17 


946 


628 


382 


211 


120 


16 


.71990 


.74674 


.77428 


.80258 


.83169 


15 


.72034 


719 


475 


306 


218 


14 


078 


764 


521 


354 


268 


13 


122 


810 


568 


402 


317 


13 


167 


855 


615 


450 


366 


11 


.72211 


.74900 


.77661 


.80498 


.83415 


10 


255 


946 


708 


546 


465 


9 


299 


991 


754 


594 


514 


S 




.75037 


801 


642 


564 


7 


388 


082 


848 


690 


613 


6 


.72432 


.75128 


.77895 


.80738 


.83662 


6 


477 


173 


941 


786 


712 


4 


521 


219 


988 


834 


761 


3 


565 


264 


.78035 


882 


811 


3 


610 


310 


082 


930 


860 


1 


.72654 


.75355 


.78129 


.80978 


.83910 





54° 


63° 


53° 


51° 


50° 


Cot 


125° 


126° 


127° 


128° 


129° 





247 



V. NATURAL TANGENTS 





139<^ 


138° 


137° 


136° 


135° 


Tan 


40° 


41° 


43° 


43° 


44° 


0' 


.83910 


.86929 


.90040 


.93252 


.96569 


1 


960 


980 


093 


306 


625 


3 


.84009 


.87031 


146 


360 


681 


3 


059 


082 


199 


415 


738 


4 


108 


133 


251 


469 


794 


5 


.84158 


.87184 


.90304 


.93524 


.96850 


6 


208 


236 


357 


678 


907 


7 


• 258 


287 


410 


633 


963 


8 


307 


338 


463 


688 


.97020 


9 


357 


389 


516 


742 


076 


10 


.84407 


.87441 


.90569 


.93797 


.97133 


11 


457 


492 


621 


852 


189 


13 


507 


543 


674 


906 


246 


13 


556 


595 


727 


961 


302 


14 


606 


646 


781 


.94016 


359 


15 


.84656 


.87698 


.90834 


.94071 


.97416 


16 


706 


749 


887 


125 


472 


17 


756 


801 


940 


180 


529 


18 


806 


852 


993 


235 


586 


19 


856 


904 


.91046 


290 


643 


30 


.84906 


.87955 


.91099 


.94345 


.97700 


21 


956 


.88007 


153 


400 


756 


33 


.85006 


059 


206 


455 


813 


33 


057 


110 


259 


510 


870 


34 


107 


162 


313 


565 


927 


35 


.85157 


.88214 


.91366 


.94620 


.97984 


36 


207 


265 


419 


676 


.98041 


37 


257 


317 


473 


731 


098 


38 


308 


369 


526 


786 


155 


39 


358 


421 


580 


841 


213 


30 


.85408 


.88473 


.91633 


.94896 


.98270 


31 


458 


524 


687 


952 


327 


33 


509 


576 


740 


.95007 


384 


33 


559 


628 


794 


062 


441 


34 


609 


680 


847 


118 


499 


35 


.85660 


.88732 


.91901 


.95173 


.98556 


36 


710 


784 


955 


229 


613 


37 


761 


836 


.92008 


284 


671 


38 


811 


888 


062 


340 


728 


39 


862 


940 


116 


395 


786 


40 


.85912 


.88992 


.92170 


.95451 


.98843 


41 


963 


.89045 


224 


506 


901 


43 


.86014 


097 


277 


562 


958 


43 


064 


149 


331 


618 


.99016 


44 


115 


201 


385 


673 


073 


45 


.86166 


.89253 


.92439 


.95729 


.99131 


46 


216 


306 


493 


785 


189 


47 


267 


358 


547 


841 


247 


48 


318 


410 


601 


897 


304 


49 


368 


463 


655 


952 


362 


50 


.86419 


.89515 


.92709 


.96008 


.99420 


51 


470 


567 


763 


064 


478 


53 


521 


620 


817 


120 


536 


63 


572 


672 


872 


176 


594 


54 


623 


725 


926 


232 


652 


55 


.86674 


.89777 


.92980 


.96288 


.99710 


56 


725 


830 


.93034 


344 


768 


57 


776 


883 


088 


400 


826 


58 


827 


935 


143 


457 


884 


59 


878 


988 


197 


513 


942 


60 


.86929 


.90040 


.93252 


.96569 


1.00000 




49° 


48° 


47° 


46° 


45° 


Cot 


130° 


13^° 


132° 


133° 


134° 



248 



AND COTANGENTS 



134° 


133° 


1 132° 


131° 


130° 


Tan 


45° 


46° 


47° 


48° 


49° 




1.00000 


1.03553 


1.07237 


1.11061 


1.15037 


6(y 


0058 


3613 


7299 


1126 


5104 


59 


0116 


3674 


7362 


1191 


5172 


58 


0175 


3734 


7425 


1256 


5240 


57 


0233 


3794 


7487 


1321 


5308 


56 


1.00291 


1.03855 


1.07550 


1.11387 


1.15375 


55 


0350 


3915 


7613 


1452 


5443 


54 


0408 


3976 


7676 


1517 


5511 


53 


0467 


4036 


7738 


1582 


5579 


53 


0525 


4097 


7801 


1648 


5647 


51 


1.00583 


1.04158 


1.07864 


1.11713 


1.15715 


50 


0642 


. 4218 


7927 


1778 


5783 


49 


0701 


4279 


7990 


1844 


5851 


48 


0759 


4340 


8053 


1909 


5919 


47 


0818 


4401 


8116 


1975 


5987 


46 


1 .00876 


1.04461 


1.08179 


1.12041 


1.16056 


45 


0935 


4522 


8243 


2106 


6124 


44 


0994 


4583 




2172 


6192 


43 


1053 


4644 


8369 


2238 


6261 


43 


1112 


4705 


8432 


2303 


6329 


41 


1.01170 


1.04766 


1.08496 


1.12369 


1.16398 


40 


1229 


4827 


8559 


2435 


6466 


39 


1288 


4888 


8622 


2501 


6535 


38 


1347 


4949 


8686 


2567 


6603 


37 


1406 


5010 


8749 


2633 


6672 


36 


1.01465 


1.05072 


1.08813 


1.12699 


1.16741 


35 


1524 


5133 


8876 


2765 


6809 


34 


1583 


5194 


8940 


2831 


6878 


33 


1642 


5255 


9003 


2897 


6947 


33 


1702 


5317 


9067 


2963 


7016 


31 


1.01761 


1 .05378 


1.09131 


1.13029 


1.17085 


30 


1820 


5439 


9195 




7154 


29 


1879 


5501 




3162 


7223 


28 


1939 


5562 


9322 


3228 


7292 


27 


1998 


5624 


9386 


3295 


7361 


26 


1 .02057 


1.05685 


1.09450 


1.13361 


1.17430 


25 


2117 


5747 


9514 


3428 


7500 


24 


2176 


5809 


9578 


3494 


7569 


23 


2236 


5870 


9642 


3561 


7638 


23 


2295 


5932 


9706 


3627 


7708 


21 


1 .02355 


1 .05994 


1.09770 


1.13694 


1.17777 


20 


2414 


6056 


9834 


3761 


7846 


19 


2474 


6117 


9899 




7916 


18 


2533 


6179 


9963 


3894 


7986 


17 


2593 


6241 


1.10027 


3961 


8055 


16 


1 .02653 


1.06303 


1.10091 


1.14028 


1.18125 


15 


2713 


6365 


0156 


4095 


8194 


14 


2772 


6427 


0220 


4162 


8264 


13 


2832 


6489 


0285 


4229 


8334 


12 


2892 


6551 


0349 


4296 


8404 


11 


1.02952 


1.06613 


1.10414 


1.14363 


1.18474 


10 


3012 


6676 


0478 


4430 


8544 


9 


3072 


6738 


0543 


4498 


8614 


8 


3132 


6800 


0607 


4565 


8684 


7 


3192 


6862 


0672 


4632 


8754 


6 


1.03252 


1.06925 


1.10737 


1.14699 


1.18824 


5 


3312 


6987 


0802 


4767 


8894 


4 


3372 


7049 


0867 


4834 


8964 




3433 


7112 


0931 


4902 


9035 


3 


3493 


7174 


0996 


4969 


9105 


1 


1.03553 


1.07237 


1.11061 


1.15037 


1.19175 





44° 


43° 


42° 


41° 


40° 


Cot 


135° 


136° 


137° 


138° 


139° 





249 







V. NATURAL TANGENTS 




129° 


128°"" 


127° 


126° 


125° 


Tan 


50° 


5i° 


53° 


53° 


54° 


0' 


1.19175 


1.23490 


1.27994 


1.32704 


1.37638 i 


1 


9246 


3563 


8071 


2785 


7722 ! 


2 


9316 


3637 


8148 


2865 


7807 


3 


9387 


S710 


8225 


2946 


7891 1 


4 


9457 


3784 


8302 


3026 


7976 ! 


5 


1.19528 


1.23858 


1.28379 


1.33107 


1.38060 


6 


9599 


S931 


8456 


3187 


8145 


I 


9669 


4005 


8533 


3268 


8229 


8 


9740 


4079 


8610 


3349 


8314 i 


9 


9811 


4153 


8687 


3430 


8399 


10 


1.19882 


1.24227 


1.28764 


1.33511 


1.38484 : 


11 


9953 


4301 


8842 


3592 


8568 i 


13 


1.20024 


4375 


8919 


3673 


8653 1 


13 


0095 


4449 


8997 


3754 


8738 


14 


0166 


4523 


9074 


3835 


8824 ' 


15 


1.20237 


1.24597 


1.29152 


1.33916 


1.38909 


16 


0308 


4672 


9229 


3998 


8994 : 


H 


0379 


4746 


9307 


4079 


9079 i 
9165 


18 


0451 


4820 


9385 


4160 


19 


0522 


4895 


9463 


4242 


9250 ; 


20 


1.20593 


1.24969 


1.29541 


1.34323 


1.39336 : 


21 


0665 


5044 


9618 


4405 


9421 


22 


0736 


5118 


9696 


4487 


9507 


23 


0808 


5193 


9775 


4568 


9593 


24 


0879 


5268 


9853 


4650 


9679 ' 


25 


1.20951 


1 .25343 


1.29931 


1.34732 


1.39764 , 


26 


1023 


5417 


1.30009 


4814 


9850 ' 


37 


1094 


5492 


0087 


4896 


9936 I 


28 


1166 


6567 


0166 


4978 


1.40022 ' 


29 


1238 


5642 


0244 


5060 


0109 


30 


1.21310 


1.25717 


1.30323 


1.35142 


1.40195 


31 


1382 


5792 


0401 


5224 


0281 1 


32 


1454 


5867 


0480 


6307 


0367 : 


33 


1526 


5943 


0558 


5389 


0454 1 


34 


1598 


6018 


0637 


6472 


0540 ^ 


35 


1.21670 


1.26093 


1.30716 


1.35554 


1.40627 ! 


36 


1742 


6169 


0795 


5637 


0714 


37 


1814 


6244 


0873 


6719 


0800 


38 


1886 


6319 


0952 


6802 


0887 


39 


1959 


6395 


1031 


5885 


0974 , 


40 


1.22031 


1.26471 


1.31110 


1.35968 


1.41061 


41 


2104 


6546 


1190 


6051 


1148 i 


42 


2176 


6622 


1269 


6134 


1235 


43 


2249 


6698 


1348 


6217 


1322 ! 


44 


2321 


6774 


1427 


6300 


1409 


45 


1.22394 


1.26849 


1.31507 


1.36383 


1.41497 


46 


2467 


6925 


1586 


6466 


1584 


47 


2539 


7001 


1666 


6549 


1672 ! 


48 


2612 


7077 


1745 


6633 


1759 


49 


2685 


7153 


1825 


6716 


1847 ! 


50 


1.22758 


1.27230 


1.31904 


1.36800 


1.41934 


51 


2831 


7306 


1984 


6883 


2022 


52 


2904 


7382 


2064 


6967 


2110 


53 


2977 


7458 


2144 


7050 


2198 ' 


54 


3050 


7535 


2224 


7134 


2286 


55 


1.23123 


1.27611 


1.32304 


1.37218 


1.42374 1 


56 


3196 


7688 


2384 


7302 


2462 


57 


3270 


7764 


2464 


7386 


2550 i 


58 


3343 


7841 


2544 


7470 


2638 


59 


3416 


7917 


2624 


7554 


2726 


60 


1.23490 


1.27994 


1.32704 


1.37638 


1.42815 j 




39° 


38° 


37° 


36° 


35° ' 


Cot 


140° 


141° 


142° 


143° 


144° 1 , 


250 1 



AND COTANGENTS 



124° 


123° 


122° 


121° 


120° 


Tan 


55° 


56° 


57° 


58° 


59° 




1.42815 


1.48256 


1.53986 


1.60033 


1.66428 


60' 


2903 


8349 


4085 


0137 


6538 


59 


2992 


8442 


4183 


0241 


6647 


5S 


3080 


8536 


4281 


0345 


6757 


57 


3169 


8629 


4379 


0449 


6867 


56 


1.43258 


1.48722 


1.54478 


1.60553 


1.66978 


55 


3347 


8816 


4576 


0657 


7088 


54 


3436 


8909 


4675 


0761 


7198 


53 


3525 


9003 


4774 


0865 


7309 


53 


3614 


9097 


4873 


0970 


7419 


51 


1.43703 


1.49190 


1.54972 


1.61074 


1.67530 


50 


3792 


9284 


5071 


1179 


7641 


49 


3881 


9378 


5170 


1283 


7752 


48 


3970 


9472 


5269 


1388 


7863 


47 


4060 


9566 


5368 


1493 


7974 


46 


.44149 


1.49661 


1.55467 


1.61598 


1.68085 


45 


4239 


9755 


6567 


1703 


8196 


44 


4329 


9849 


5666 


1808 


8308 


43 


4418 


9944 


6766 


1914 


8419 


43 


4508 


1.50038 




2019 


8531 


41 


1.44598 


1.50133 


1.55966 


1.62125 


1.68643 


40 


4688 


0228 


6065 


2230 


8754 


39 


4778 




6165 


2336 


8866 


38 


4868 


0417 


6265 


2442 


8979 


37 


4958 


0512 


6366 


2548 


9091 


36 


1.45049 


1.50607 


1.56466 


1.62654 


1.69203 


35 


5139 


0702 


6566 


2760 


9316 


34 


5229 


0797 


6667 


2866 


9428 


33 


5320 


0893 


6767 


2972 


9541 


33 


5410 


0988 


6868 


3079 


9653 


31 


1.45501 


1.51084 


1.56969 


1.63185 


1.69766 


30 


5592 


1179 


7069 


3292 


9879 


39 


6682 


1275 


7170 


3398 


9992 


38 


5773 


1370 


7271 


3505 


1.70106 


37 


6864 


1466 


7372 


3612 


0219 


36 


1.45955 


1.51562 


1.57474 


1.63719 


1.70332 


35 


6046 


1658 


7575 




0446 


34 


6137 


1754 


7676 


3934 


0560 


33 


6229 


1850 


7778 


4041 


0673 


33 


6320 


1946 


7879 


4148 


0787 


31 


1.46411 


1.52043 


1.57981 


1.64256 


1.70901 


30 


6503 


2139 


8083 


4363 


1015 


19 


6595 


2235 


8184 


4471 


1129 


18 


6686 


2332 


8286 


4579 


1244 


17 


6778 


2429 


8388 


4687 


1358 


16 


1.46870 


1.52525 


1.58490 


1.64795 


1.71473 


15 


6962 


2622 


8593 


4903 


1588 


14 


7053 


2719 


8695 


5011 


1702 


13 


7146 


2816 


8797 


5120 


1817 


13 


7238 


2913 


8900 


5228 


1932 


11 


1.47330 


1.53010 


1.59002 


1.65337 


1.72047 


10 


7422 


3107 


9105 


5445 


2163 


9 


7514 


3205 


9208 


5554 


2278 


8 


7607 


3302 


9311 


5663 


2393 


7 


7699 


3400 


9414 


5772 


2509 


6 


1.47792 


1.53497 


1.59517 


1.65881 


1.72625 


5 


7885 


3595 


9620 


6990 


2741 


4 


7977 


3693 


9723 


6099 


2857 


3 


8070 


3791 


9826 


6209 


2973 


3 


8163 


3888 


9930 


6318 


3089 


1 


1.48256 


1.53986 


1.60033 


1.66428 


1.73205 





34° 


33° 


32° 


31° 


30° 


Cot 


145° 


146° 


147° 


148° 


149° 





251 



V. NATURAL TANGENTS 





119° 


118° 


117° 1 116° 


115° 


Tan 


60° 


61° 


62° 


63° 


64° 


0' 


1.73205 


1 .80405 


1 88073 


1.96261 


2.05030 


1 


3321 


0529 


8205 


6402 


6182 


2 


3438 


0653 


8337 


6544 


6333 




3555 


0777 


8469 


6685 


6485 


^ 


3671 


0901 


8602 


6827 


5637 


5 


1.73788 


1.81025 


1.88734 


1.96969 


2.05790 


6 


3905 


1150 


8867 


7111 


6942 




4022 


1274 


9000 


7253 


6094 


s 


4140 


1399 


9133 


7395 


6247 


d 


4257 


1524 


9266 


7538 


6400 


10 


1.74375 


1.81649 


1.89400 


1.97681 


2.06553 


11 


4492 


1774 


9533 


7823 


6706 


13 


4610 


1899 


9667 


7966 


6860 


13 


4728 


2025 


9801 


8110 


7014 


14 


4846 


2150 


9935 


8253 


7167 


15 


1.74964 


1.82276 


1.90069 


1.98396 


2.07321 


16 


5082 


2402 


0203 


8540 


7476 


17 


5200 


2528 


0337 


8684 


7630 


18 


6319 


2654 


0472 


8828 


7785 


19 


5437 


2780 


0607 


8972 


7939 


20 


1.75556 


1.82906 


1.90741 


1.99116 


2.08094 


31 


6675 


3033 


0876 


9261 


8250 


33 


5794 


3159 


1012 


9406 


8405 


33 


5913 


3286 


1147 


9550 


8560 


34 


6032 


3413 


1282 


9695 


8716 


35 


1.76151 


1.83540 


1.91418 


1.99841 


2.08872 


36 


6271 


3667 


1554 


9986 


9028 


37 


6390 


3794 




2.00131 


9184 


38 


6510 


3922 


1826 


0277 


9341 


39 


6629 


4049 


1962 


0423 


9498 


30 


1.76749 


1.84177 


1.92098 


2.00569 


2.09654 


31 


6869 


4305 


2235 


0715 


9811 


33 


6990 


4433 


2371 


0862 


9969 


33 


7110 


4561 


2508 


1008 


2.10126 




7230 


4689 


2645 


1155 


0284 


35 


1.77351 


1.84818 


1.92782 


2.01302 


2.10442 


36 


7471 


4946 


2920 


1449 


0600 


37 


7592 


5075 


3057 


. 1596 


0758 
0916 


38 


7713 


5204 


3195 


1743 


39 


7834 


5333 


3332 


1891 


1075 


40 


1.77955 


1.85462 


1.93470 


2.02039 


2.11233 


41 


8077 


5591 


3608 


2187 


1392 


43 


8198 


5720 


3746 


2335 


1552 


43 


8319 


5850 


3885 


2483 


1711 


44 


8441 


5979 


4023 


2631 


1871 


45 


1.78563 


1.86109 


1.94162 


2.02780 


2.12030 


46 


8685 


6239 


4301 


2929 


2190 


47 


8807 


6369 


4440 


3078 


2350 


48 


8929 


6499 


4579 


3227 


2511 


49 


9051 


6630 


4718 


3376 


2671 


60 


1.79174 


1.86760 


1.94858 


2.03526 


2.12832 


61 


9296 


6891 


4997 


3675 


2993 


63 


9419 


7021 


5137 


3825 


3154 


63 


9542 


7152 


5277 


3975 


3316 


64 


9665 


7283 


5417 


4125 


3477 I 


65 


1.79788 


1.87415 


1.95557 


2.04276 


2.13639 


66 


9911 


7546 


5698 


4426 


3801 


67 


1.80034 


7677 


5838 


4577 


3963 


68 


0158 


7809 


5979 


4728 


4125 


69 


0281 


7941 


6120 


4879 


4288 ! 


60 


1.80405 


1.88073 


1.96261 


2.05030 


2.14451 




29° 


28° 


27° 


26° 


25° 


Cot 


150° 


151° 


152° 


153° 


154° 1 


252 1 



AND COTANGENTS 



114° 


113° 


112° 


111° 


110° 


Tan 


65° 


66° 


67° 


68° 


69° 




2.14451 


2.24604 


2.35585 


2.47509 


2.60509 


60' 


4614 


4780 


5776 


7716 


0736 


59 


4in 


4956 


5967 


7924 


0963 


58 


4940 


5132 


6158 


8132 


1190 


57 


5104 


5309 


6349 


8340 


1418 


56 


2.15268 


2.25486 


2.36541 


2.48549 


2.61646 


55 


5432 


5663 


6733 


8758 


1874 


54 


5596 


5840 


6925 


8967 


2103 


53 


5760 


6018 


7118 


9177 


2332 


53 


5925 


6196 


7311 


9386 


2561 


51 


2.16090 


2.26374 


2.37504 


2.49597 


2.62791 


50 


6255 


6552 


7697 


9807 


3021 


49 


6420 


6730 


7891 


2.50018 


3252 


48 


6585 


6909 


8084 


0229 


3483 


47 


6751 


7088 


8279 


0440 


3714 


46 


2.16917 


2.27267 


2.38473 


2.50652 


2.63945 


45 


7083 


7447 


8668 


0864 


4177 


44 


724-9 


7626 


8863 


1076 


4410 


43 


7416 


7806 


9058 


1289 


4642 


43 


7582 


7987 


9253 


1502 


4875 


41 


2.17749 


2.28167 


2.39449 


2.51715 


2.65109 


40 


7916 


8348 


9645 


1929 


5342 


39 


8084 


8528 


9841 


2142 


5576 


38 


8251 


8710 


2.40038 


2357 


5811 


37 


8419 


8891 


0235 


2571 


6046 


36 


2.18587 


2.29073 


2.40432 


2.52786 


2.66281 


35 


8755 


9254 


0629 


3001 


6516 


34 


8923 


9437 


0827 


3217 


6752 


33 


9092 


9619 


1025 


3432 


6989 


33 


9261 


9801 


1223 


3648 


7225 


31 


2.19430 


2.29984 


2.41421 


2.53865 


2.67462 


30 


9599 


2.30167 


1620 


4082 


7700 


39 


9769 


0351 


1819 


4299 


7937 


38 


9938 


0534 


2019 


4516 


8175 


37 


2.20108 


0718 


2218 


4734 


8414 


36 


2.20278 


2.30902 


2.42418 


2.54952 


2.68653 


35 


0449 


1086 


2618 


5170 


8892 


34 


0619 


1271 


2819 


5389 


9131 


33 


0790 


1456 


3019 


5608 


9371 


33 


0961 


1641 


3220 


5827 


9612 


31 


2.21132 


2.31826 


2.43422 


2.56046 


2.69853 


30 


1304 


2012 


3623 


6266 


2.70094 


19 


1475 


2197 


3825 


6487 


0335 


18 


1647 


2383 


4027 


6707 


0577 


17 


1819 


2570 


4230 


6928 


0819 


16 


2.21992 


2.32756 


2.44433 


2.57150 


2.71062 


15 


2164 


2943 


4636 


7371 


1305 


14 


2337 


3130 


4839 


7593 


1548 


13 


2510 


3317 


5043 


7815 


1792 


13 


« 2683 


3505 


5246 


8038 


2036 


11 


2.22857 


2.33693 


2.45451 


2.58261 


2.72281 


10 


3030 


3881 


5655 


■ 8484 


2526 


9 


3204 


4069 


5860 


8708 


2771 


8 


3378 


4258 


6065 


8932 


3017 


7 


3553 


4447 


6270 


9156 


3263 


6 


; 2.23727 


2.34636 


2.46476 


2.59381 


2.73509 


5 


3902 


4825 


6682 


9606 


3756 


4 


4077 


5015 


6888 


9831 


4004 


3 


i 4252 


5205 


7095 


2.60057 


4251 


3 


4428 


5395 


7302 


0283 


4499 


1 


2-24804 


2.35585 


2.47509 


2.60509 


2.74748 





. 34° 


23° 


23° 


31° 


20° 


Cot 


' 155° 


156° 


157° 


158° 


159° 





253 







V. 


NATURAL TANGENTS | 




109° 


108° 


107° 


106° 


105° I 


Tan 


70° 


71° 


72° 


73° 


74° 1 


0' 


2.74748 


2.90421 


3.07768 


3.27085 


3.48741 1 


1 


74997 


90696 


08073 


27426 


49125 li 


2 


75246 


90971 


08379 


27767 


49509 


3 


75496 


91246 


08685 


28109 


49894 


4 


75746 


91523 


08991 


28452 


60279 


5 


2.75996 


2.91799 


3.09298 


3.28795 


3.60666 


6 


76247 


92076 


09606 


29139 


61053 


7 


76498 


92354 


09914 


29483 


61441 


8 


76750 


92632 


10223 


29829 


61829 


9 


77002 


92910 


10532 


30174 


62219 


10 


2.77254 


2.93189 


3.10842 


3.30521 


3.52609 


11 


77507 


93468 


11153 


30868 


53001 


13 


77761 


93748 


11464 


31216 


63393 


13 


7B014 


94028 


11775 


31565 


63785 


14 


78269 


94309 


12087 


31914 


64179 ' 


15 


2.78523 


2.94591 


3.12400 


3.32264 


3.54573 


16 


78778 


94872 


12713 


32614 


54968 > 


17 


79033 


95155 


13027 


32965 


65364 1 


IS 


79289 


95437 


13341 


33317 


65761 


19 


79545 


95721 


13656 


33670 


66159 




2.79802 


2.96004 


3.13972 


3.34023 


3.56557 


21 


80059 


96288 


14288 


34377 


56957 


23 


80316 


96573 


14605 


34732 


57357 


23 


80574 


96858 


14922 


S5087 


67758 


24 


80833 


97144 


15240 


35443 


58160 


25 


2.81091 


2.97430 


3.15558 


3.35800 


3.58562 


26 


81350 


97717 


15877 


36158 


58966 




81610 


98004 


16197 


36516 


59370 i 


28 


81870 


98292 


16517 


36875 


59775 


29 


82130 


98580 


16838 


37234 


60181 , 


30 


2.82391 


2.9886S 


3.17159 


3.37594 


3.60588 ; 


31 


82653 


99158 


17481 


37955 


60996 


32 


82914 


99447 


17804 


38317 


61405 


33 


83176 


99738 


18127 


38679 


61814 


34 


83439 


3.00028 


18451 


39042 


62224 


35 


2.83702 


3.00319 


3.18775 


3.39406 


3.62636 


36 


83965 


00611 


19100 


39771 


63048 


37 


84229 


00903 


19426 


40136 


63461 


38 


84494 


01196 


19752 


40502 


63874 


39 


84758 


01489 


20079 


40869 


64289 


40 


2.85023 


3.01783 


3.20406 


3.41236 


3.64705 


41 


85289 


02077 


20734 


41604 


65121 


42 


85555 


02372 


21063 


41973 


65538 


43 


85822 


02667 


21392 


42343 


65957 


44 


86089 


02963 


21722 


42713 


66376 


45 


2.86356 


3.03260 


3.22053 


3.43084 


3.66796 


46 


86624 


03556 


22384 


43456 


67217 


47 


86892 


03854 


22715 


43829 


67638 


48 


87161 


04152 


23048 


44202 


68061 


49 


87430 


04450 


23381 


44576 


68485 


50 


2.87700 


3.04749 


3.23714 


3.44951 


3.68909 


61 


87970 


05049 


24049 


45327 


69335 


53 


88240 


05349 


24383 


45703 


69761 


53 


88511 


05649 


24719 


460S0 


70188 


54 


88783 


05950 


25055 


46458 


70016 


55 


2.89055 


3.06252 


3.25392 


3.46837 


3.71046 i 


56 




06554 


25729 


47216 


71476 


67 


89600 


06857 


26067 


47596 


71907 


68 


89873 


07160 


26406 


47977 


72338 


59 


90147 


07464 


26745 


48359 


72771 ' 


60 


2.90421 


3.07768 


3.27085 

r^° 


3.48741 


3.73205 




19° 


1S° 


16° 


3 5° 


Cot 


160° 


161° 


162° 


163° 


164° 






2. 


54 




i 
1 



mD COTANGENTS 



1040 


■"1030 


1020 


101° 


100<> 


" Tan ■ 


TS*'"^ 


76° 


77° 


78° 


79° 




3.73205 


4.01078 


4.33148 


4.70463 


5.14455 


eo'"' 


73640 


01576 


33723 


71137 


15256 


69 


74075 


02074 


34300 


71813 


16058 


68 


74512 


02574 


34879 


72490 


16863 


67 


74950 


03076 


35459 


73170 


17671 


56 


8.75388 


4.03578 


4.36040 


4.73851 


5.18480 


55 


75828 


04081 


3662^1 


74534 


19293 


54 


76268 


04586 


37207 


75219 


20107 


63 


76709 


05092 


37793 


75906 


20925 


53 


77152 


05599 




76593 


21744 


51 


3.77595 


4.06107 


4.38969 


4.77286 


5.22566 


50 


78040 


06616 


S9560 


77978 


23391 


49 


78485 


07127 


40152 


78673 


24218 


48 


78931 


07639 


40745 


79370 


25048 


47 


79378 


08152 


41340 


80068 


25880 


46 


3.79827 


4.08666 


4.41936 


4.80769 


5.26715 


46 


80276 


09182 


42534 


81471 


27553 


44 


80726 


09699 


43134 


82175 


28393 


43 


81177 


10216 


43735 


82882 


29235 


43 


81630 


10736 


44338 


83590 


30080 


41 


3.82083 


4.11256 


4.44942 


4.84300 


5.30928 


40 


82537 


11778 


45548 


85013 


31778 


39 


82992 


12301 


46155 


85727 


32631 


38 


83449 


12825 


46764 


86444 


33487 


37 


83906 


13350 


47374 


87162 


34345 


36 


3.84364 


4.13877 


4.47986 


4.87882 


5.35206 


35 


84824 


14405 


48600 


88605 


36070 


34 


85284 


14934 


49215 


89330 


36936 


33 


85745 


15465 


49832 


90056 


37805 


33 


86208 


15997 


60451 


90785 


38677 


31 


3.86671 


4.16530 


1.51071 


4,91516 


5.39552 


30 


87136 


17064 


51693 


92249 


40429 


39 


87601 


17600 


52316 


92984 


41309 


38 


88068 


18137 


52941 


93721 


42192 


37 


88536 


18675 


53568 


94460 


43077 


36 


3.89004 


4.19215 


4.54196 


4.95201 


5.43966 


35 


89474 


19756 


.54826 


95945 


44857 


34 


89945 


20298 


55458 


96690 


45751 


33 


90417 


20842 


56091 


97438 


46648 


33 


90890 


21387 


56726 


98188 


47548 


31 


3.91364 


4.21933 


4.57363 


4.98940 


5.48451 


30 


91839 


22481 


58001 


99695 


49356 


19 


92316 


23030 


68641 


5.00451 


50264 


18 


92793 


23580 


69283 


01210 


61176 


17 


93271 


24132 


59927 


01971 


52090 


16 


3 .93751 


4.24685 


4.60572 


5.02734 


5.53007 


15 


94232 


25239 


61219 


03499 


53927 


14 


94713 


25795 


61868 


04267 


64851 


13 


95196 


26352 


62518 


05037 


65777 


13 


95680 


26911 


63171 


05809 


56706 


11 


3.96165 


4.27471 


4 .63825 


5.06584 


5.57638 


10 


96651 


28032 


64480 


07360 


68573 


9 


97139 


28595 


65138 


08139 


59511 


8 


97627 


29159 


65797 


08921 


60452 


7 


98117 


29724 


66458 


09704 


61397 


6 


3.98607 


4 .30291 


4.67121 


5.10490 


5.62344 


6 


99099 


30860 


67786 


11279 


63295 


4 


99592 


31430 


68452 


12069 


64248 


3 


4 .00086 


32001 


69121 


12862 


65205 


2 


00582 


32573 


69791 


13658 


66165 


1 


4 .01078 


4.33148 


4.70463 


5.14455 


5.67128 





1 140 


13° 


12° 


11° 


10° 


Cot 


1 165° 


166° 


167° 


168° 


169° 





255 



V. NATURAL TANGENTS 





99° 


98° 


97° 


96° 


95° 


Tan 


80° 


81° 


83° 


83° 


84° 


0' 


5.67128 


6.31375 


7.11537 


8.14435 


9.51436 


1 


68094 


32566 


13042 


16398 


54106 


2 


69064 


33761 


14553 


18370 


66791 


3 


70037 


34961 


16071 


20352 


59490 


4 


71013 


36165 


17594 


22344 


62205 


5 


5.71992 


6.37374 


7.19125 


8.24345 


9.64935 


6 


72974 


38587 


20661 


26355 


67680 


7 


73960 


39804 


22204 


28376 


70441 


8 


74949 


41026 


23754 


30406 


73217 


9 


75941 


42253 


25310 


32446 


76009 


10 


5.76937 


6.43484 


7.26873 


8.34496 


9.78817 


11 


77936 


44720 


28442 


36555 


81641 


13 


78938 


45961 


30018 


38625 


84482 


13 


79944 


47206 


31600 


40705 


87338 


14 


80953 


48456 


33190 


42795 


90211 


15 


5.81966 


6.49710 


7.34786 


8.44896 


9.93101 


16 


82982 


50970 


36389 


47007 


96007 


17 


84001 


52234 


37999 


49128 


98931 


18 


85024 


53503 


39616 


51259 


10.0187 


19 


86051 


54777 


41240 


53402 


0.0483 


30 


5.87080 


6.56055 


7.42871 


8.55555 


10.0780 


31 


88114 


57339 


44509 


57718 


0.1080 


33 


89151 


58627 


46154 


59893 


0.1381 


33 


90191 


59921 


47806 


62078 


0.1683 


34 


91236 


61219 


49465 


64275 


0.1988 


35 


5.92283 


6.62523 


7.51132 


8.66482 


10.2294 


36 


93335 


63831 


52806 


68701 


0.2602 


37 


94390 


65144 


54487 


70931 


0.2913 


38 


95448 


66463 


56176 


73172 


0.3224 


39 


96510 


67787 


57872 


75425 


0.3538 


30 


5.97576 


6.69116 


7.59575 


8.77689 


10.3854 


31 


98646 


70450 


61287 


79964 


0.4172 


33 


99720 


71789 


63005 


82252 


0.4491 


33 


6.00797 


73133 


64732 


84551 


0.4813 


34 


01878 


74483 


66466 


86862 


0.5136 


35 


6.02962 


6.75838 


7.68208 


8.89185 


10.5462 


36 


04051 


77199 


69957 


91520 


0.5789 


37 


05143 


78564 


71715 


93867 


0.6118 


38 


06240 


79936 


73480 


96227 


0.6450 


39 


07340 


81312 


75254 


98598 


0.6783 


40 


6.08444 


6.82694 


'' '78825 


9.00983 


10.7119 


41 


09552 


84082 




03379 


0.7457 


43 


10664 


85475 


80622 


05789 


0.7797 


43 


11779 


86874 


82428 


08211 


0.8139 


44 


12899 


88278 


84242 


10646 


0.8483 


45 


6.14023 


6.89688 


7.86064 


9.13093 


10.8829 


46 


15151 


91104 


87895 


15554 


0.9178 


47 


16283 


92525 


89734 


18028 


0.9529 


48 


17419 


93952 


91582 


20516 


0.9882 


49 


18559 


95385 


93438 


23016 


1 .0237 


50 


6.19703 


6.96823 


7.95302 


9.25530 


11.0594 


51 


20851 


98268 


97176 


28058 


1.0954 


53 


22003 


99718 


99058 


30599 


1.1316 


53 


23160 


7.01174 


8.00948 


33155 


1.1681 


54 


24321 


02637 


02848 


35724 


1.2048 


55 


6.25486 


7.04105 


8.04756 


9.38307 


11.2417 


56 


26655 


05579 


06674 


40904 


1.2789 


57 


27829 


07059 


08600 


43515 


1.3163 


58 


29007 


08546 


10536 


46141 


1.3540 


59 


30189 


10038 


12481 


48781 


1.3919 


60 


6.31375 


7.11537 


8.14435 


9.51436 


11.4301 




9° 


8° 


7° . 


6° 


5° 


Cot 


170° 


171° 


172° 


173° 


174° 



256 



AND COTANGENTS 



94° 


93° 


92° 


91° 


90° 


Tan 


85° 


86° 


87° 


88° 


89° 




11.4301 


14.3007 


19.0811 


28.6363 


57.2900 


6(y 


1.4685 


4.3607 


9.1879 


8.8771 


8.2612 


59 


1.5072 


4.4212 


9.2959 


9.1220 


9.2659 


58 


1.5461 


4.4823 


9.4051 


9.3711 


60.3058 


57 


1.5853 


4.5438 


9.5156 


9.6245 


1.3829 


56 


11.6248 


14.6059 


19.6273 


29.8823 


62.4992 


55 


1.6645 


4.6685 


9.7403 


30.1446 


3.6567 


54 


1.7045 


4.7317 


9.8546 


0.4116 


4.8580 


53 


1.7448 


4.7954 


9.9702 


0.6833 


6.1055 


52 


1.7853 


4.8596 


20.0872 


0.9599 


7.4019 


51 


11.8262 


14 .9244 


20.2056 


31.2416 


68.7501 


50 


1.8673 


4.9898 


0.3253 


1.5284 


70.1533 


49 


1 .9087 


5.0557 


0.4465 


1.8205 


1.6151 


48 


1.9504 


5.1222 


0.5691 


2.1181 


3.1390 


47 


1.9923 


5.1893 


0.6932 


2.4213 


4.7292 


46 


12.0346 


15.2571 


20.8188 


32.7303 


76.3900 


45 


2.0772 


5.3254 


0.9460 


3.0452 


8.1263 


44 


2.1201 


5.3943 


1.0747 


3.3662 


9.9434 


43 


2.1632 


5.4638 


1.2049 


3.6935 


81.8470 


42 


2.2067 


5.5340 


1.3369 


4.0273 


3 8435 


41 


12.2505 


15.6048 


21.4704 


34.3678 


85.9398 


40 


2.2946 


5.6762 


1.6056 


4.7151 


8.1436 


39 


2.3390 


5.7483 


1.7426 


5.0695 


90.4633 


38 




5.8211 


1.8813 


5.4313 


2.9085 




2*4288 


5.8945 


2.0217 


5.8006 


5.4895 


36 


12!4742 


15.9687 


22.1640 


36.1776 


98.2179 


35 


2.5199 


6.0435 


2.3081 


6.5627 


101.107 


34 


2.5660 


6.1190 


2.4541 


6.9560 


04.171 


33 


2.6124 


6.1952 


2.6020 


7.3579 


07.426 


32 


2.6591 


6.2722 


2.7519 


7.7686 


10.892 


31 


12.7062 


16.3499 


22.9038 


38.1885 


114.589 


30 


2.7536 


6.4283 


3.0577 


8.6177 


18.540 


29 


2.8014 


6.5075 


3.2137 


9.0568 


22.774 


28 


2.8496 


6.5874 


3.3718 


9.5059 


27.321 


27 


2.8981 


6.6681 


3.5321 


9.9655 


32.219 


26 


12.9469 


16.7496 


23.6945 


40.4358 


137.507 


25 


2.9962 


6.8319 


3.8593 


0.9174 


43.237 


24 


3.0458 


6.9150 


4.0263 


1.4106 


49.465 


23 


3.0958 


6.9990 


4.1957 


1.9158 


56.259 


22 


3.1461 


7.0837 


4.3675 


2.4335 


63.700 


21 


13.1969 


17.1693 


24.5418 


42.9641 


171.885 


20 


3.2480 


7.2558 


4.7185 


3.5081 


80.932 


19 


3.2996 


7.3432 


4.8978 


4.0661 


90.984 


18 


3.3515 


7.4314 


5.0798 


4.6386 


202.219 


17 


3.4039 


7.5205 


5.2644 


5.2261 


14 .858 


16 


13.4566 


17.6106 


25.4517 


45.8294 


229.182 


15 


3.5098 


7.7015 


5.6418 


6.4489 


45.552 


14 


3.5634 


7.7934 


5.8348 


7.0853 


64.441 


13 


3.6174 


7.8863 


6.0307 


7.7395 


86.478 


12 


3.6719 


7.9802 


6.2296 


8.4121 


312,521 


11 


13.7267 


18.0750 


26.4316 


49.1039 


343.774 


10 


3.7821 


8.1708 


6.6367 


9.8157 


381 .971 


9 


3.8378 


8.2677 


6.8450 


50.5485 


429.718 


S 


3.8940 


8.3655 


7.0566 


1.3032 


491.106 


7 


3.9507 


8.4645 


7.2715 


2.0807 


572.957 


6 


14.0079 


18.5645 


27.4899 


52.8821 


687.549 


5 


4 .0655 


8.6656 


7.7117 


3.7086 


859.436 


4 


4.1235 


8.7678 


7.9372 


4.5613 


1145.92 


3 


4.1821 


8.8711 


8.1664 


5.4415 


1718.87 


2 


4.2411 


8.9755 


8.3994 


6.3506 


3437.75 


1 


14.3007 


19.0811 


28.6363 


57.2900 


Infinite 





4° 


3° 


2° 


1° 


0° 


Cot 


1750 


176° 


177° 


178° 


179° 





J57 



258 TABLES 

VI. CONVERSION FACTORS. 
Angles. 

1 rad.= 57.2958 deg.= 3437.75 miii. = 206,265 sec. 

Areas. 

1 sq. mile = 640 acres = 258.999 hectares. 

1 hectare = 100 ares =» 10,000 sq. meters = 2.471 

acres. 
1 acre =10 sq. chains = 43,560 sq. ft. 
1 sq. yd. = 9 sq. ft. = 0.836 sq. meter. 
1 sq. meter= 10.764 sq. ft. = 1.196 sq. yd. 

Densities. 

1 lb. per cu. ft. = 16.018 kg. per cu. meter. 
1 lb. per cu. in. = 27.680 g. per cu. cm. 
1 kg. per cu. meter =0.06243 lb. per cu. ft. 
1 g. per cu. cm. = 0.03613 lb. per cu. in. 

Discharge. 

1 cu. ft. per sec. = 448.9 gal. per min. =1.9835 

acre-ft. per day. 
1 acre-ft. per day = 0.5042 cu. ft. per sec. 
1,000,000 gal. per day = 3.0689 acre-ft. per day 

= 1.547 ca. ft. per sec. 
1 cu. ft. per sec. = 40 miner's inches. 
1 miner's inch = 1.5 cu. ft. per min. = 11.22 gal. 

per min. 
1 in. of rainfall per hr. = 1.008 cu. ft. per sec. 

per acre. 

Energy. 

1 ft-lb. = 1.356 joules or watt-sec. 

1 joule = 10^ ergs= 10^ dyne-cm. 

1 horse-power-hr. = 1.98X 10^ ft-lb. = 0.7457 kw- 

hr. = 2544 Btu. 
1 kw-hr. = 1.341 horse-power-hr. = 3411 Btu. = 

2.654X106 ft-Ib. 
1 Btu. = 778.4 ft-lb. = 0.252 kg-cal. 
1 meter-kilogram = 7.233 ft-lb. 



TABLES 259 



Force. 

1 lb. = 0.4536 kg. = 444,822 dynes. 
1 kg. = 2.2046 lb. = 980,665 dynes. 
1,000,000 dynes = 2.2481 lb. = 1.020 kg. 

Length. 

1 mile = 5280 ft. = 80 chains = 320 rods =1.6094 

kilometers. 
1 meter = 39.37 inches = 3.2808 ft. = 1.0936 yd. 
1 in. = 2.54 cm. = 25.4 mm. 
1 yd. = 0.9144 meter. 
1 ft. = 30.48 cm. = 0.3048 meter. 

Power. 

1 horse-power =33,000 ft-lb. per min. = 550 ft- 

Ib. per sec. 
1 horse-power = 0.7457 kw. = 0.7066 Btu. per 

sec. 
1 kw. = 1.341 horse-power = 737.5 ft-lb. per sec. 
1 horse-power =1.0139 metric horse-power. 

Pressure. 

1 ft. of water =62.4 lb. per sq. ft. = 0.433 lb. per 

sq. in. 
1 in. of mercury = 1.134 ft. of water = 0.4912 lb. 

per sq. in. 
1 atmosphere =14.697 lb. per sq. in. = 33.9 ft. 

of water. 
1 lb. per sq. ft. = 4.8824 kg. per sq. meter. 
1 lb. per sq. in. = 0.07031 kg. per sq. cm. 
1 kg. per sq. cm. = 14.223 lb. per sq. in. = 32.8 

ft. of water. 
1 ton per sq. ft. = 13.889 lb. per sq. in. 

Temperature. 

Deg. C.= (deg. F.- 32) X 0.55556. 
Deg. F. = (1.8Xdeg. C.)+32. 



260 TABLES 

Velocity. 

1 rad. per sec. = 9.5496 rev. per min. = 0.15916 

rev. per sec. 
1 rev. per min. = 6.0000 deg. per sec. 
1 ft. per sec. = 0.6818 miles per hr. 
1 mile per hr. = 88 ft. per min. = 1.4667 ft. per sec. 

Volume. 

1 cu. yd. = 27 cu. ft. = 21.696 bushels. 

1 cu. meter = 1000 liters = 1.308 cu. yds. 

1 bu. = 8 gal. (dry) = 1.2445 cu. ft. = 2150.4 cu. 

in. 
1 gal. (dry measure) = 1.1637 gal. (liquid 

measure) . 
1 cu. ft. = 7.481 gal. (liquid measure). 

Weight. 

1 lb. Avoir. = 1.2153 lb. Troy or Apoth. 

1 lb. Avoir. = 16 oz. = 7000 grains = 0.4536 kg. 

1 kg. = 2.2046 lb. Avoir. 

1 short ton = 2000 lb. = 0.90718 metric ton. 

1 long ton =2240 lb. = 1.120 short tons. 

1 metric ton =1000 kg. = 2204.6 lb. 



^ 







Vn. PROPERTIES OF 1 


2 ^ >> 


. 


1«.' 


Thermal Head 


-u 




1^1 






in B.t.u. 






gJl-H © 


of 


of 


Ph ^ 




>-^ 


liquid. 


vapor. 






P 


t 


V" 


i' 


i" 


r 


1 


79.1 


652 


47.1 


1095.0 


1047.9 


1.2 


84.7 


549 


52.7 


97.6 


44.9 




1.4 


89.5 


474.3 


57.6 


99.8 


42.3 




1.6 


93.8 


418.2 


61.8 


1101.8 


40.0 




1.8 


97.7 


374.3 


65.7 


03.5 


37.9 




2 


101.2 


338.9 


69.2 


1105.1 


1036.0 




3 


115.1 


231.4 


83.0 


11.4 


28.3 




4 


123.4 


176.5 


93.4 


15.9 


22.5 




6 


140.8 


120.7 


108.7 


22.6 


13.9 




8 


152.3 


92.1 


120.2 


27.5 


07.4 




10 


161.5 


74.8 


129.4 


1131.4 


1002.1 




15 


179.1 


51.1 


147.0 


38.8 


991.7 




20 


192.4 


39.1 


160.3 


44.1 


83.8 


1 ! 


25 


203.1 


31.7 


170.1 


48.3 


77.3 




29.92 


212.0 


26.8 


180.0 


51.7 


71,7 




lb. per 




■ 










sq. in. 














15 


213.0 


26.30 


181.0 


1152.2 


971.2 


;• 


16 


216.3 


24.76 


184.3 


53.4 


69.1 




17 


219.4 


23.40 


187.5 


54.6 


67.1 




18 


222.4 


22.18 


190.5 


55.7 


65.2 




19 


225.2 


21.09 


193.3 


56.7 


63.4 




20 


228.0 


20.10 


196.0 


1157.7 


961.7 




22 


233.1 


18.38 


201.2 


59.6 


58.4 




24 


237.8 


16.95 


206.0 


61.3 


55.3 




26 


242.2 


15.73 


210.4 


62.8 


52.4 




28 


246.4 


14.67 


214.6 


64.3 


49.7 




30 


250.3 


13.76 


218.6 


1165.7 


947.1 




32 


254.0 


12.95 


222.4 


• 66.9 


44.6 




34 


257.6 


12.24 


225.9 


68.1 


42.2 


j 


36 


260.9 


11.60 


229.4 


69.2 


39.9 




38 


264.2 


11.03 


232.6 


70.3 


37.7 




40 


267.2 


10.51 


235.8 


1171.3 


935.5 




42 


270.2 


10.04 


238.8 


72.2 


33.5 




44 


273.0 


9.61 


241.7 


73.2 


31.5 




46 


275.8 


9.22 


244.5 


74.0 


29.6 




48 


278.4 


8.86 


247.2 


74.8 


27.7 




50 


281.0 


8.53 


249.8 


1175.6 


925.9 




52 


283.5 


8.22 


252.3 


76.4 


24.1 




54 


285.9 


7.93 


254.7 


77.1 


22.4 




56 


288.2 


7.67 


257.1 


77.8 


20.7 




58 


290.5 


7.42 


259.5 


78.5 


19.0 




60 


292.7 


7.18 


261.7 


1179.1 


917.4 


( 


62 


294.9 


6.97 


263.9 


79.7 


15.8 




64 


296.9 


6.76 


266.1 


80.3 


14.3 




66 


299.0 


6.57 


268.2 


80.9 


12.7 




68 


301.0 


6.39 


270.2 


81.5 


11.2 








262 






i 



SATURATED STEAM (Goodenough). 



Energy in 
B.t.u. 


Entropy 


^ o 3 


of vapor- 


of 


. ^{ 


of vapor- 


of 


0--I 


ization. 


vapor. 


liquid. 


ization. 


vapor. 


Oh ^ 


P 


u" 


s' 


r 
T 


s" 


P 


988.7 


1035.8 


0.0915 


1.9455 


2.0370 


1 


85.0 


37.7 


.1019 


.9198 


.0217 


1.2 


81.9 


39.4 


.1108 


.8980 


.0087 


1.4 


79.1 


40.9 


.1185 


.8791 


1.9976 


1.6 


76.6 


42.3 


.1254 


.8624 


.9878 


1.8 


974.3 


1043.5 


0.1316 


1.8474 


1.9790 


2 


65.2 


48.2 


.1561 


.7893 


.9454 


3 


58.3 


51.7 


.1739 


.7478 


.9217 


4 


48.1 


56.8 


.1998 


.6888 


.8886 


6 


40.4 


60.5 


.2187 


.6464 


.8651 


8 


934.1 


1063.5 


0.2336 


1.6134 


1.8470 


10 


22.0 


69.0 


.2617 


.5526 


.8143 


15 


12.7 


73.1 


.2822 


.5089 


.7912 


20 


05.2 


76.2 


.2986 


.4747 


.7733 


25 


898.8 


78.8 


.3120 


.4469 


.7589 


29.92 
lb. per 
sq. in. 


898.1 


1079.1 


0.3135 


1.4438 


1.7573 


15 


95.8 


80.0 


.3184 


.4337 


.7521 


16 


93.5 


80.9 


.3230 


.4242 


.7473 


17 


91.4 


81.7 


.3274 


.4153 


.7427 


18 


89.3 


82.5 


.3316 


.4068 


.7384 


19 


887.3 


1083.3 


0.3356 


1.3987 


1.7343 


20 


83.6 


84.7 


.3430 


.3837 


.7267 


22 


80.1 


85.9 


.3499 


.3698 


.7197 


24 


•76.8 


87.1 


.3563 


.3570 


.7133 


26 


73.7 


88.2 


.3622 


.3452 


.7074 


28 


870.7 


1089.2 


0.3679 


1.3340 


1.7019 


30 


67.9 


90.2 


.3731 


.3236 


.6967 


32 


65.2 


91.0 


.3781 


.3137 


.6918 


34 


62.7 


91.9 


.3829 


.3044 


.6873 


36 


60.2 


92.7 


.3874 


.2956 


.6830 


38 


857.8 


1093.4 


0.3917 


1.2871 


1.6788 


40 


55.5 


94.2 


.3958 


.2791 


.6749 


42 


53.3 


94.8 


.3998 


.2714 


.6712 


44 


51.2 


95.5 


.4086 


.2640 


.6676 


46 


49.1 


96.1 


.4072 


.2570 


.6642 


48 


847.1 


1096.7 


0.4108 


1.2501 


1.6609 


50 


45.1 


97.2 


.4142 


.2436 


.6577 


52 


43.2 


97.8 


.4174 


.2373 


.6547 


54 


41.4 


98.3 


.4206 


.2311 


.6517 


56 


39.5 


98.8 


.4237 


.2252 


.6489 


58 


837.8 


1099.3 


0.4267 


1.2195 


1.6462 


60 


36.0 


99.7 


.4296 


.2139 


.6435 


62 


34.3 


1100.2 


.4324 


.2085 


.6409 


64 


31.1 


01.0 


.4379 


.1981 


.6360 


66 


32.7 


00.6 


.4352 


.2032 


.6384 


68 



263 



Vn. PROPERTIES OF 



bod 



70 

72 

74 

76 

78 

80 

82 

84 

86 

88 

90 

92 

94 

96 

98 

100 

105 

110 

115 

120 

125 

130 

135 

140 

145 

150 

155 

160 

165 

170 

175 

180 

185 

190 

195 

200 

210 

220 

230 

240 

250 

260 

270 

280 

300 



C2< • 



302. 

304. 

306.7 

308.5 

310.3 

312.0 

313.7 

315.4 

317.1 

318.7 

320.3 

321.8 

323.3 

324.8 

326.3 

327.8 

331.4 

334.8 

338.1 

341.3 

344.4 

347.4 

350.3 

353.1 

355 

358.5 

361 

363.6 

366 

368.5 

370.8 

373.1 

375.4 

377.6 

379.7 

381.9 

386.0 

390.0 

393.8 

397.5 

401.1 

404.5 

407.9 

411.2 

417.5 






6.22 

6.05 

5.90 

5.75 

5.61 

5.48 

5.35 

5.23 

5.12 

5.01 

4.905 

4.805 

4.709 

4.617 

4.528 

4.442 

4.240 

4.057 

3.889 

3.735 

3.593 

3.461 

3.340 

3.226 

3.120 

3.020 

2.927 

2.839 

2.757 

2.679 

2.605 

2.536 

2.470 

2.408 

2.348 

2.292 

2.186 

2.090 

2.002 

1.921 

1.846 

1.777 

1.713 

1.654 

1.545 



Thermal Head 
in B.t.u. 



of 
liquid. 



272.2 

274.2 

276.1 

278.0 

279.8 

281.6 

283.4 

285.1 

286.8 

288.5 

290.1 

291.7 

293.3 

294.8 

296.4 

297.9 

301.6 

305.1 

308.6 

311.9 

315.1 

318.2 

321.2 

324.2 

327.0 

329.8 

332.5 

335.2 

337.8 

340.3 

342.8 

345.2 

347.6 

350.0 

352.2 

354.5 

358.8 

363.0 

367.1 

371.0 

374.9 

378.6 

382.2 

385.7 

392.4 



of 
vapor. 



1182.0 
82.5 
83.0 
83.5 
84 

1184 
84 
85 
85 
86.1 

1186.5 
86.9 
87.3 
87.7 
88.0 

1188.4 
89.2 
90.0 
90.7 
91.4 

1192.0 
92.6 
93.2 
93.7 
94.2 

1194.7 
95.2 
95.7 

■ 96.1 
96.5 

1196.9 
97.2 
97.6 
97.9 



98 

1198 
99 
99. 
99 

1200 

1200, 
01.0 
01.2 
01.5 
01.9 



^1^ s 
4S © -M 



2M 



SATURATED 


STEAM 


(Goodenough). 




Energy in 
B.t.u. 


Entropy 


Pressure, 
Lb. per 
Sq. In. 


of vapor- 
ization. 


of 
vapor. 


of 
liquid. 


of vapor- 
ization. 


of 
vapor. 


P 


u" 


s' 


r 
T 


s" 


P 


829.5 


1101.4 


0.4405 


1.1931 


1.6336 


70 


27.9 


01.8 


.4431 


.1883 


.6313 


72 


26.4 


02.2 


.4456 


.1835 


.6291 


74 


•24.9 


02.6 


.4480 


.1789 


.6269 


76 


23.4 


02.9 


.4504 


.1744 


.6248 


78 


821.9 


1103.2 


0.4527 


1.1700 


1.6227 


80 


20.5 


03.6 


.4550 


.1657 


.6207 


82 


19.1 


03.9 


.4572 


.1615 


.6187 


84 


17.7 


04.2 


.4594 


.1574 


.6168 


86 


16.3 


04.5 


.4615 


.1534 


.6149 


88 


815.0 


1104.8 


0.4636 


1.1495 


1.6131 


90 


13.7 


05.1 


.4657 


.1456 


.6113 


92 


12.4 


05.4 


.4677 


.1419 


.6096 


94 


11.1 


05.6 


.4697 


.1381 


.6079 


96 


09.8 


05.9 


.4717 


.1345 


.6062 


98 


808.6 


1106.2 


0.4736 


1.1309 


1.6045 


100 


05.5 


06.8 


.4782 


.1222 


.6004 


105 


02.6 


07.3 


.4827 


.1138 


.5965 


110 


799.7 


07.9 


.4870 


.1058 


.5928 


115 


96.9 


08.4 


.4911 


.0982 


.5893 


120 


794.2 


1108.8 


0.4950 


1.0908 


1.5858 


125 


91.6 


09.3 


.4989 


.0836 


.5825 


130 


89.0 


09.7 


.5026 


.0767 


.6793 


135 


86.4 


10.1 


.5062 


.0700 


.5762 


140 


84.0 


10.5 


.5097 


.0636 


.5733 


145 


781.6 


1110.9 


0.5131 


1.0573 


1.5704 


150 


79.2 


11.2 


.5164 


.0512 


.5676 


155 


76.9 


11.5 


.5196 


.0453 


.5649 


160 


74.6 


11.8 


.5227 


.0395 


.5622 


165 


72.4 


12.1 


.5258 


.0339 


.5597 


170 


770.2 


1112.4 


0.5287 


1.0284 


1.5572 


175 


68.0 


12.7 


.5316 


.0231 


.5547 


180 


65.9 


12.9 


.5344 


.0179 


.5523 


185 


63.9 


13.2 


.5372 


.0128 


.5500 


190 


61.8 


13.4 


.5399 


.0079 


.5478 


195 


759.8 


1113.6 


0.5426 


1.0030 


1.5456 


200 


55.9 


14.0 


.5477 


.9936 


.5413 


210 


52.1 


14.3 


.5526 


.9846 


.5372 


220 


48.3 


14.6 


.5573 


.9760 


.5333 


230 


44.7 


14.9 


.5619 


.9676 


.5295 


240 


741.2 


1115.2 


0.5663 


0.9595 


1.5258 


250 


37.7 


15.4 


.5706 


.9517 


.5223 


260 


34.4 


15.6 


.5747 


.9442 


.5189 


270 


31.1 


15.8 


.5787 


.9369 


.5156 


280 


24.7 


16.0 


.5863 


.9229 


.5092 


300 



265 



Vm. PRESSURE-ENTROPY TABLE 



|§3« 


1.50 


1.55 1 




X 


i 


V 


X 


i 


V 


300 


0.990 


1194 


1.53 


474 


1239 


1.70 


280 


0.983 


1188 


1.63 


458 


1232 


1.80 


260 


0.977 


1182 


1.74 


442 


1225 


1.90 


240 


0.970 


1175 


1.86 


425 


1218 


2.02 


220 


0.962 


1168 


2.01 


407 


1210 


2.16 


200 


0.955 


1160 


2.19 


388 


1202 


2.32 


190 


0.951 
0.947 


1156 
1152 


2.29 
2.40 


378 


1198 


2.41 


180 


0.995 


1193 


2.52 


170 


0.942 


1147 


2.53 


0.991 


1188 


2.65 


160 


0.938 


1142 


2.66 


0.986 


1183 


2.80 


150 


0.933 


1137 


2.82 


0.981 


1178 


2.96 


140 


0.929 


1132 


3.00 


0.976 


1172 


3.15 


130 


0.924 


1126 


3.20 


0.970 


1166 


3.36 


120 


0.919 


1120 


3.43 


0.964 


1160 


3.60 


110 


0.913 


1113 


3.71 


0.958 


1153 


3.89 


100 


0.908 


1106 


4.03 


0.952 


1145 


4.23 


95 


0.905 


1102 


4.22 


0.949 


1141 


4.42 


90 


0.902 


1098 


4.42 


0.945 


1137 


4.64 


85 


0.898 


1094 


4.65 


0.942 


1133 


4.87 


80 


0.895 


1090 


4.90 


0.938 


1128 


5.13 


76 


0.892 


1086 


5.13 


0.935 


1124 


5.37 


72 


0.890 


1082 


5.38 


0.932 


1120 


5.64 


68 


0.887 


1078 


5.67 


0.928 


1116 


5.93 


64 


0.883 


1074 


5.97 


0.925 


1112 


6.25 


60 


0.880 


1069 


6.32 


0.921 


1107 


6.62 


56 


0.877 


1064 


6.73 


0.917 


1102 


7.04 


52 


0.873 


1059 


7.18 


0.913 


1096 


7.51 


48 


0.869 


1054 


7.70 


0.909 


1090 


8.05 


44 


0.865 


1048 


8.32 


0.905 


1084 


8.70 


40 


0.861 


1041 


9.05 


0.900 


1077 


9.46 


36 


0.856 


1034 


9.93 


0.895 


1070 


10.38 


32 


0.851 


1027 


11.03 


0.889 


1062 


11.52 


28 


0.846 


1018 


12.42 


0.883 


1053 


12.97 


24 


0.840 


1008 


14.23 


0.876 


1043 


14.85 


20 


0.833 


997 


16.73 


0.868 


1031 


17.45 


18 


0.829 


990 


18.38 


0.864 


1024 


19.17 


16 

In.Hg 

30 


0.824 


983 


20.41 


0.859 


1017 


21.27 


0.821 


978 


21.96 


0.856 


1012 


22.88 


24 


0.813 


965 


26.80 


0.847 


998 


27.91 


20 


0.807 


954 


31. 5i 


0.840 


987 


32.84 


16 


0.800 


942 


38.51 


0.832 


974 


40.06 


12 


0.791 


926 


49.8 


0.822 


957 


51.83 


10 


0.785 


916 


58.7 


0.816 


947 


61.0 


8 


0.778 


904 


71.7 


0.809 


935 


74.5 


6 


0.770 


889 


92.9 


0.800 


919 


96.5 


5 


0.765 


880 


109.5 


0.794 


910 


113.7 


4 


0.759 


869 


133.9 


0.787 


898 


138.9 


3 


0.751 


855 


173.8 


0.779 


884 


180.3 


2 


0.741 


837 


251.0 


0.768 


865 


260.2 


1 


0.724 


806 


472 


0.750 


833 


489 



266 



FOR STEAM 


(Goodenoiigh). 






Pressure 
Lb. per 
Sq. In. 


1.60 


1.65 


X 


{ 


V 


z 


^ 


V 


300 


554 


1287 


1.91 


648 


1340 


2.13 


280 


538 


1280 


2.01 


630 


1332 


2.25 


260 


520 


1272 


2.13 


612 


1324 


2.38 


■ 240 


502 


1264 


2.26 


592 


1315 


2.53 


220 


483 


1256 


2.42 


571 


1305 


2.70 


200 


462 


1246 


2.60 


548 


1295 


2.91 


190 


450 


1241 


2.70 


536 


1289 


3.02 


180 


439 


1236 


2.82 


524 


1283 


3.15 


170 


427 


1231 


2.94 


511 


1277 


3.29 


160 


414 


1226 


3.08 


497 


1271 


3.44 


150 


401 


1220 


3.23 


482 


1264 


3.62 


140 


387 


1213 


3.41 


467 


1258 


3.81 


130 


372 


1207 


3.60 


451 


1251 


4.03 


120 


356 


1200 


3.83 


434 


1243 


4.28 


110 
100 


340 


1193 


4.09 


416 
396 


1235 
1226 


4.58 


0.996 


1185 


4.42 


4.92 


95 


0.992 


1181 


4.63 


386 


1221 


5.12 


90 


0.988 


1176 


4.85 


375 


1216 


5.33 


85 


0.985 


1172 


5.10 


364 


1211 


5.57 


80 


0.981 


1167 


5.37 


352 


1206 


5.83 


76 


0.977 


1163 


5.62 


342 


1202 


6.07 


72 


0.974 


1159 


.5.89 


332 


1197 


6.32 


68 


0.970 


1154 


6.20 


321 


1192 


6.59 


64 


0.966 


1149 


6.53 


310 


1187 


6.90 


60 
56 


0.962 
0.958 


1144 
1139 


6.91 
7.35 


298 


1182 


7.24 


0.999 


1177 


7.66 


52 


0.954 


1134 


7.84 


0.994 


1171 


8.17 


48 


0.949 


1128 


8.41 


0.989 


1164 


8.76 


44 


0.944 


1121 


9.08 


0.983 


1158 


9.45 


40 


0.939 


1114 


9.87 


0.978 


1150 


10.28 


36 


0.933 


1106 


10.82 


0.971 


1142 


11.26 


32 


0.927 


1098 


12.01 


0.965 


1134 


12.50 


28 


0.920 


1089 


13.51 


0.957 


1124 


14.06 


24 


0.913 


1078 


15.47 


0.949 


1113 


16.09 


20 


0.904 


1065 


18.17 


0.940 


1100 


18.89 


18 


0.899 


1058 


19.95 


0.935 


1092 


20.73 


16 


0.894 


1051 


22.14 


0.929 


1084 


23.00 


In.Hg 














30 


0.890 


1045 


23.81 


0.925 


1079 


24.73 


24 


0.881 


1031 


29.03 


0.915 


1064 


30.14 


20 


0.873 


1019 


34.13 


0.907 


1052 


35.43 


16 


0.864 


1006 


41.62 


0.897 


1038 


43.18 


12 


0.854 


989 


53.8 


0.885 


1020 


55.8 


10 


0.847 


978 


63.3 


0.878 


1009 


65.7 


8 


0.839 


965 


77.3 


0.869 


996 


80.1 


6 


0.829 


949 


100.1 


0.859 


979 


103.6 


5 


0.823 


939 


117.9 


0.852 


969 


122.1 


4 


0.816 


928 


144.0 


0.845 


957 


149.0 


3 


0.807 


913 


186.7 


0.835 


942 


193.2 


2 


0.795 


893 


269 


0.822 


921 


279 


1 


0.775 


869 


506 


0.801 


887 


522 



267 



vm. 


PRESSURE-ENTROPY TABLE 


sure 
per 
In. 




1.70 






1.75 
















Press 
Lb. 
Sq. 


X 


i 


V 


X 


i 


V 


150 


577 
560 
543 
524 


1314 
1306 
1298 
1290 


4.03 
4.25 
4.50 

4.78 








140 








130 








120 


"m 


iui' 


"'b.Z2 


110 


504 


1281 


5.11 


604 


1331 


5.69 


100 


483 


1271 


5.50 


581 


1320 


6.12 


96 


474 


1267 


5.67 


571 


1316 


6.32 


92 


464 


1263 


5.86 


561 


1311 


6.52 


88 


455 


1258 


6.06 


550 


1306 


6.75 


84 


445 


1253 


6.27 


539 


1301 


6.99 


80 


434 


1240 


6.51 


528 


1296 


7.26 


76 


423 


1244 


6.77 


516 


1290 


7.55 


72 


412 


1239 


7.06 


504 


1284 


7.87 


68 


400 


1233 


7.37 


491 


1278 


8.22 


64 


388 


1228 


7.72 


478 


1272 


8.61 


60 


375 


1222 


8.11 


463 


1266 


9.04 


58 


368 


1219 


8.32 


456 


1262 


9.28 


56 


361 


1216 


8.54 


448 


1259 


9.53 


54 


354 


1212 


8.78 


441 


1255 


9.80 


52 


347 


1209 


9.03 


433 


1252 


10.08 


50 


340 


1206 


9.30 


424 


1248 


10.38 


48 


332 


1202 


9.59 


416 


1244 


10.71 


46 


324 


1199 


9.91 


407 


1240 


11.06 


44 


316 


1195 


10.25 


398 


1236 


11.45 


42 


307 


1191 


10.61 


388 


1231 


11.86 


40 


298 


1187 


11.01 


379 


1227 


12.31 


38 


289 


1183 


11.45 


368 


1222 


12.80 


36 


279 


1179 


11.93 


358 


1217 


13.33 


34 


269 


1174 


12.46 


347 


1212 


13.93 


32 


258 


1169 


13.04 


335 
323 


1207 
1202 


14.58 
15.31 


30 


0.999 


1164 


13.74 


28 


0.995 


1159 


14.60 


310 


1196 


16.13 


26 


0.990 


1154 


15.58 


296 


1190 


17.06 


24 


0.986 


1148 


16.71 


282 


1183 


18.12 


22 


0.981 


1141 


18.03 


267 


1176 


19.37 


20 


0.976 


1134 


19.61 


250 


1169 


20.81 


18 


0.970 
0.964 


1127 
1118 


21.52 
23.86 


233 


1161 


22.54 


16 


0.999 


1152 


24.73 


In.Hg 














30 


0.960 


1112 


25.66 


0.994 


1146 


26.58 


24 


0.948 


1097 


31.25 


0.982 


1130 


32.36 


20 


0.940 


1085 


36.72 


0.973 


1117 


38.02 


16 


0.929 


1070 


44.74 


0.962 


1102 


46.30 


12 


0.917 


1052 


57.78 


0.948 


1083 


59.77 


10 


0.909 


1040 


68.0 


0.940 


1071 


70.3 


8 


0.900 


1027 


82.9 


0.930 


1057 


85.7 


6 


0.888 


1009 


107.2 


0.918 


1039 


110.7 


5 


0.881 


999 


126.2 


0.911 


1029 


130.4 


4 


0.873 


986 


154.1 


0.902 


1015 


159.1 


3 


0.863 


970 


200 


0.891 


999 


206 


2 


0.849 


949 


288 


0.876 


977 


297 


1 


0.827 


914 


539 


0.853 


940 


556 



268 



FOR STEAM (Goodenough). 



isd 




1.80 






1.85 




TO '="•-' 


























2^ 6' 


z 


i 


V 


X 


i 


V 


90 

88 
86 
84 
82 
80 
78 
76 
74 
72 
70 
68 
66 
64 
62 
60 
58 
56 
54 
52 
50 


663 
657 
651 
645 
639 
633 
627 
620 
614 
607 
600 
593 
585 
578 
571 
563 
555 
546 
538 
529 
520 


1362 
1359 
1356 
1354 
1351 
1348 
1345 
1341 
1338 
1335 
1332 
1329 
1325 
1322 
1318 
1314 
1311 
1307 
1303 
1299 
1294 


7.38 

7.50 

7.64 

7.78 

7.92 

8.08 

8.24 

8.40 

8.57 

8.75 

8.94 

9.14 

9.35 

9.58 

9.82 

10.06 

10.33 

10.61 

10.91 

11.23 

11.57 


























































































































626 


1346 


12.87 


48 


511 


1290 


11.93 


616 


1341 


13.28 


46 


501 


1286 


12.33 


605 


1336 


13.72 


44 


491 


1281 


12.76 


595 


1331 


14.20 


42 


480 


1276 


13.22 


583 


1326 


14.71 


40 


470 


1271 


13.72 


572 


1320 


15.27 


38 


458 


1266 


14.26 


559 


1314 


15.89 


36 


447 


1260 


14.86 


546 


1308 


16.56 


34 


434 


1255 


15.52 


533 


1302 


17.30 


32 


422 


1249 


16.26 


519 


1295 


18.12 


30 


408 


1243 


17.08 


504 


1288 


19.03 


28 


394 


1236 


18.00 


488 


1281 


20.05 


26 


379 


1230 


19.04 


472 


1274 


21.23 


24 


363 


1222 


20.23 


455 


1266 


22.55 


22 


346 


1214 


21.62 


436 


1257 


24.10 


20 


328 


1206 


23.23 


416 


1248 


25.92 


18 


308 


1197 


25.17 


394 


1238 


28.08 


16 


287 


1187 


27.52 


370 


1227 


30.69 


In. Hg 














30 


272 


1181 


29.34 


354 


1219 


32.73 


24 


234 


1163 


34.78 


312 


1200 


38.81 


20 
16 


204 


1150 


40.0 


279 
240 


1185 
1167 


44.6 


0.994 


1134 


47.9 


52.9 


12 


0.980 


1114 


61.8 


193 


1146 


65.9 


10 

8 


0.971 
0.961 


1102 
1088 


72.6 

88.5 


166 


1133 


75.3 


0.991 


1118 


91.3 


6 


0.948 


1069 


114.3 


0.977 


1099 


117.9 


5 


0.940 


1058 


134.6 


0.969 


1088 


138.7 


4 


0.930 


1045 


164.2 


0.959 


1074 


169.2 


3 


0.919 


1028 


212.6 


0.947 


1057 


219.1 


2 


0.903 


1005 


306 


0.930 


1033 


315 


1 


0.878 


967 


573 


0.904 


994 


589 



269 



INDEX. 

Acceleration 46, 47 

Adiabatic of Gases 105 

Admittance 145 

Admittance of Parallel Circuits .... 144 

Air Compression 119 

Alternating Current 140 

Alternating Current Power 141 

Alternating Current Power Factor . . . 142 

Analytic Geometry 14 

Apparent Power 143, 145 

Areas (See Circle, Ellipse, etc.). 

Arithmetical Progression 3 

Bazin's Formula (Chezy Coefficient) . . 100 

Bazin's Formula (Weirs) 93 

Beams 65 

Cantilever Beam, End Load 73 

Cantilever Beam, Uniform Load ... 72 

Continuous Beams 77, 78 

Fixed Beam, Center Load 74 

Fixed Beam, Load off Center .... 76 

Fixed Beam, Uniform Load 73 

Reinforced Concrete Beams 87 

Simple Beam, Center Load 70 

Simple Beam, Load off Center ... 70 

Simple Beam, Moving Loads 71 

I Simple Beam, Several Loads 71 

Simple Beam, Uniform Load .... 69 

Theorem of Three Moments 67 

Bending Moment 66 

Belt, Friction of 53 

Bernoulli's Theorem 99 

Binomial Theorem 4 

Boiler Horsepower 117 

Boilers, Steam 116 

Brayton Cycle 117 

Calculus, Differential 23 

271 



272 INDEX 

PAGE 

Calculus, lotegral 26 

Capacitance, Capacity 131 

Capacitance of Cable 132 

Capacitance of Transmission Line . . . 132 

Catenary, The 21 

Center of Gravity 41 

Center of Pressure 93 

Centroid of Forces 41 

Charging Current 133 

Chezy's Formula 98, 99 

Circle of Inertia 45 

Circle, The 17 

Columns 80 

Columns, Reinforced Concrete 87 

Combined Stresses 84 

Combinations and Permutations .... 3 

Compound Air Compressors 120 

Concrete, Reinforced 86 

Conductance 145 

Conductance of Parallel Circuits . . . 144 

Condensers 131, 132 

Cones 21 

Conic Sections, General Equation of . . 22 

Conversion Factors 258 

Core Sections 59 

Couple, Definition 41 

Cubic Equations, Solutions for 5 

Curves (See Analytic Geometry) . . . 17-22 

Curves, Elastic 68 

Cycloid, The 20 

Cylinders, Thin, Stresses in 63 

Deflection of a Beam 69 

Delta Connection 147 

Density of Water 92 

Determinants 4 

Diesel Cycle 117 

Differential Calculus 23 

Differentiation 24 

Direct Current Dynamos: 

Efficiency of ., 139 



INDEX 273 

PAGE 

Torque of 138 

Voltage of 138 

Diverging Nozzles Ill 

Dynamics 46 

Eddy-current Loss 131 

Effective Value, E. M. F. or Current . . 141 

Efficiency of D-C. Dynamo 139 

Efficiency of Steam Engine 116 

Elastic curves 68 

Electric Circuit: 

General Equation 140 

Growth and Decay of Current in . . . 139 
Electromotive Force Magnetically In- 
duced 129 

EHipse, The 18 

Ellipsoid of Inertia 45 

Ellipsoid of Stress 86 

Energy 52 

Energy in Condenser 133 

Energy in Magnetic Field 130 

Energy of Jets 96 

Entropy Defined 103 

Equations, Methods of Solution .... 5 

Equilibrium 40 

Equivalent Impedance 144 

Equivalent Pipe Lengths 99 

Equivalent Reactance 144 

Equivalent Resistance 144 

Exponents 1 

Falling Bodies 46 

Fanning's Formula for Pipes 97 

Flamant's Formula for Pipes 98 

Flow in Channels 99 

Flow in Pipes 97 

Flow of Compressed Air 114 

Flow of Fluids 108 

Flow of Gases in Mains 113 

Flow of Steam 114 

Force and Acceleration 47 

Force of Jet 96 



I 



274 INDEX 

iAGE 

Forces 40 

Form Factor 141 

Francis' Formula for Weirs 93 

Friction 52 

Friction of Belt 63 

Fteley and Stearns' Formula for Weirs . 93 

Gas Engines 117 

Gases, Equation of 103 

Gases, Changes of State of ...... 104 

Geometrical Progression 3 

Graphical Solution of Equations .... 6 

Grashof's Formula for Flow of Steam . . 110 
Greek Letters (Facing Page 1). 

Harmonic Progression 3 

Harmonic E.M.F. or Current . . . 140, 141 

Herschel's Formula for Weirs 95 

Higher Equations, Method of Solution . 6 

Hydraulic Grade Line 101 

Hydraulics 92 

Hyperbola, The 19 

Hyperbolic Functions 13 

Hyperboloids 22 

Hysteresis Loss 131 

Impact 47,86 

Impedance 141 

Impulse and Reaction cf Water .... 96 

Inductance 129 

Inductance of Transmission Lines . . . ? 30 

Inductance, Mutual 130 

Inductive Circuit, Growth and Decay of 

Current - ^ 1^9 

Inertia Circle ^^ 

Inertia, Moment of ^^ 

Integral Calculus 2" 

Internal Combustion Engine ^^^ 

Isothermal of Gases ^^^ 

Joule Cycle ^^"^ 

Kernel of a Section ^^ 

Kirchoff's Laws 1^7 

Kutter's Formula lOP 



INDEX 275 

PAGE 

Logarithms, Theory and Explanation . . 1 

Logarithms of Numbers 156 

Logarithmic Sines and Cosines 186 

Logarithmic Tangents and Cotangents . 204 

Losses of Head, Hydraulic ...... 98 

Magnetic Circuit 128 

Magnetic Fields 127 

Magnetic Fields, Energy Stored in . . . 130 

Magnetic Flux 128 

Magnetic Forces 127 

Magnetic Permeability 128 

Magnetic Reluctance 128 

Magnetomotive Force 128 

Materials, Strength of 54 

Maximum Value of a Function » . . . 25 

Mechanics of Materials 54 

Mechanics, Theoretical 39 

Minimum Value of a Function 25 

Moment, Bending 66 

Moment of Inertia 43 

Movement for Concentrated Loads ... 71 

Moving Plates 96 

Motion of a Point, Curvilinear 48 

Napier's Rule for Flow of Steam ... 110 

Networks 137 

Neutral Axis, Equation of 59 

Nozzles Ill 

Ohm's Law 133 

Orifices and Jets 95 

Otto Cycle 117 

Parabola, The 18 

Parabola, The Cubic 21 

Paraboloids 22 

Parallel Circuits: 

Impedance of . 143 

Resistance of 135 

Solution of, Carrjdng Alternating Cur- 
rent 143, 144 

Solution of. Carrying Direct Current . 137 

Pendulum 52 



276 INDEX 

PAGE 

Permutations and Combinations .... 3 

Pipes, Flow in 97 

Pipes, Stresses in ... 63 

Plane Triangles 11 

Power 52 

Power Factor: 

Loads in Parallel 145 

Loads in Series 143 

Three-phase System 148 

Power for Loads in Series 143 

Power for Loads in Parallel 145 

Power for Three-phase Circuits 148 

Power Lost in Resistance 135 

Power, Transmission of 53, 86 

Precessional Rotation 51 

Product of Inertia 44 

Progression 3 

Projectiles 48 

Proportion 2 

Quadratic Equations 5 

Radius of Curvature 25 

Radius of Curvature for Beams .... 68 

Radius of G5a-ation 44 

Rankine Cycle 114 

Rateau's Formula for Flow of Steam . . 110 

Reactance 141 

Reactive Volt-amperes 143, 145 

Refrigeration, Air 121 

Refrigeration with Vapor 122 

Reinforced Concrete 86 

Resistance of Conductors 133, 134 

Resistance of Parallel Circuits 135 

Resistance of Series Circuits 135 

Resistivity. 134 

Riveted Joints 64 

Rotation 49 

Rotation, Precessional 51 

Section Modulus 60 

Series ••••••• 4 



INDEX 277 

PAGE 

Series Circuits: 

Impedance of 142 

Resistance of 135 

Solution of, Carrying Alternating Cur- 
rent 142, 143 

Solution of, Carrying Direct Current. . 136 

Shear, Diagonal 62 

Shear, Vertical 65 

Shearing Stresses 66 

Sines and Cosines, Logarithmic 186 

Sines and Cosines, Natural 222 

Smith's (Hamilton) Formula for Weirs . 94 

Speed of D-C. Motor 139 

Spheres 21 

Spheres, Thin, Stresses in 63 

Spherical Triangles 12 

Spheroids , . . 21 

Spirals 20 

Star Connection 146 

Static Pressure 92 

Statics 40 

Steam Boilers 116 

Steam, Changes of State of 108 

Steam Engine 114 

Steam, Fundamental Relations .... 106 

Steam Tables Explained , . 107 

Straight Line, The 16 

Stresses: 

Combined Stresses 84 

Diagonal Stresses 62 

Direct Stresses 55 

Eccentric Stresses ........ 56, 84 

Flexural Stresses 68 

Stresses in Cylinders and Spheres . . 63 

Struts 80 

Submerged Weirs 94 

Superheated Steam, Equations for . . . 107 

Thermal Head 103 

Thermodynamics, Equations of . . * 102 

Susceptance .»•.,*• 145 



278 INDEX 

PAGB 

Susceptance of Parallel Circuits .... 144 

Tangents and Cotangents, Logarithmic . 204 

Tangents and Cotangents, Natural . . . 240 

Taylor's Theorem 25 

Temperature Coefficient of Resistance. . 134 

Theorem of Three Moments 67 

Three-phase Circuits 145 

Top, Rotation of 51 

Torsion 85 

Transformation Formulae 44 

Transformation of Coordinates .... 14 

Transmission of Power 53, 86 

Triangles, Solution of 11 

Trigonometry 8 

Tube, Standard 96 

Velocity and Acceleration 46 

Velocity Head 95 

Velocities, Relative, in Channels .... 101 

Velocities, Virtual 47 

Volumes (See Sphere, Cone, etc.). 

Weight of Copper Wires 134 

Weirs 93 

Work and Energy ...•.,•••• 53 



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